Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 91.3%

Time: 10.6s

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: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-294}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))))
   (*
    c0_s
    (if (<= (* V l) (- INFINITY))
      t_0
      (if (<= (* V l) -1e-294)
        (* c0_m (/ (sqrt (- A)) (sqrt (* l (- V)))))
        (if (<= (* V l) 4e-319)
          t_0
          (if (<= (* V l) 2e+281)
            (* c0_m (/ (sqrt A) (sqrt (* V l))))
            (sqrt (* A (* (/ c0_m l) (/ c0_m V)))))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * (sqrt((A / -l)) / sqrt(-V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((V * l) <= -1e-294) {
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	} else if ((V * l) <= 4e-319) {
		tmp = t_0;
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if ((V * l) <= -1e-294) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((V * l) <= 4e-319) {
		tmp = t_0;
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0
	elif (V * l) <= -1e-294:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (V * l) <= 4e-319:
		tmp = t_0
	elif (V * l) <= 2e+281:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-294)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(V * l) <= 4e-319)
		tmp = t_0;
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * (sqrt((A / -l)) / sqrt(-V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0;
	elseif ((V * l) <= -1e-294)
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	elseif ((V * l) <= 4e-319)
		tmp = t_0;
	elseif ((V * l) <= 2e+281)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-294], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -inf.0 or -1.00000000000000002e-294 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 42.4%

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

      1. associate-/r* 72.4%

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

      2. div-inv 72.4%

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

      3. div-inv 72.4%

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

      4. associate-*l* 40.7%

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

   4. Applied egg-rr 40.7%

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

      1. frac-times 40.7%

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

      2. metadata-eval 40.7%

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

      3. div-inv 42.4%

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

      4. associate-/l/ 72.5%

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

      5. frac-2neg 72.5%

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

      6. sqrt-div 54.6%

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

      7. distribute-neg-frac2 54.6%

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

   6. Applied egg-rr 54.6%

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

      1. distribute-frac-neg2 54.6%

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

      2. distribute-frac-neg 54.6%

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

   8. Simplified 54.6%

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

   if -inf.0 < (*.f64 V l) < -1.00000000000000002e-294

   1. Initial program 88.9%

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

      1. frac-2neg 88.9%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

   if 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 19.4%

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

      1. associate-/r* 67.4%

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

      2. div-inv 67.3%

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

      3. div-inv 67.3%

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

      4. associate-*l* 25.9%

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

   4. Applied egg-rr 25.9%

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

      1. sqrt-prod 25.9%

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

      2. div-inv 25.9%

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

      3. associate-*r* 25.6%

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

      4. *-commutative 25.6%

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

      5. div-inv 25.6%

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

      6. frac-times 19.1%

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

      7. metadata-eval 19.1%

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

      8. sqrt-div 19.1%

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

      9. metadata-eval 19.1%

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

      10. div-inv 19.1%

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

      11. add-sqr-sqrt 19.1%

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

      12. sqrt-unprod 19.1%

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

      13. frac-times 18.2%

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

   6. Applied egg-rr 17.9%

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

      1. associate-/l* 18.4%

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

      2. associate-/r* 26.2%

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

   8. Simplified 26.2%

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

      1. associate-/l/ 18.4%

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

      2. unpow2 18.4%

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

      3. times-frac 46.9%

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

   10. Applied egg-rr 46.9%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-294}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+276}\right):\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 4e+276)))
      (/ c0_m (sqrt (* V (/ l A))))
      t_0))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+276)) {
		tmp = c0_m / sqrt((V * (l / A)));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+276))) then
        tmp = c0_m / sqrt((v * (l / a)))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+276)) {
		tmp = c0_m / Math.sqrt((V * (l / A)));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+276):
		tmp = c0_m / math.sqrt((V * (l / A)))
	else:
		tmp = t_0
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+276))
		tmp = Float64(c0_m / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+276)))
		tmp = c0_m / sqrt((V * (l / A)));
	else
		tmp = t_0;
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+276]], $MachinePrecision]], N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.0000000000000002e276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 67.2%

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

      1. associate-/r* 73.7%

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

      2. div-inv 73.7%

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

      3. div-inv 73.7%

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

      4. associate-*l* 67.2%

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

   4. Applied egg-rr 67.2%

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

      1. sqrt-prod 38.1%

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

      2. div-inv 38.2%

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

      3. *-commutative 38.2%

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

      4. div-inv 38.1%

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

      5. frac-times 37.6%

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

      6. metadata-eval 37.6%

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

      7. sqrt-div 38.4%

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

      8. metadata-eval 38.4%

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

      9. associate-/r/ 38.3%

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

      10. un-div-inv 38.4%

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

      11. sqrt-undiv 66.9%

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

   6. Applied egg-rr 66.9%

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

      1. associate-*r/ 72.5%

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

   8. Simplified 72.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e276

   1. Initial program 98.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+276}\right):\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 4e+231)))
      (* c0_m (sqrt (/ (/ A V) l)))
      t_0))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+231)) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+231))) then
        tmp = c0_m * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+231)) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+231):
		tmp = c0_m * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+231))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+231)))
		tmp = c0_m * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+231]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 4.0000000000000002e231 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 67.9%

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

      1. associate-/r* 74.3%

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

   3. Simplified 74.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e231

   1. Initial program 98.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+276}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (<= t_0 0.0)
      (sqrt (* A (* (/ c0_m l) (/ c0_m V))))
      (if (<= t_0 4e+276) t_0 (/ c0_m (sqrt (* V (/ l A)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	} else if (t_0 <= 4e+276) {
		tmp = t_0;
	} else {
		tmp = c0_m / sqrt((V * (l / A)));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = sqrt((a * ((c0_m / l) * (c0_m / v))))
    else if (t_0 <= 4d+276) then
        tmp = t_0
    else
        tmp = c0_m / sqrt((v * (l / a)))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	} else if (t_0 <= 4e+276) {
		tmp = t_0;
	} else {
		tmp = c0_m / Math.sqrt((V * (l / A)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	elif t_0 <= 4e+276:
		tmp = t_0
	else:
		tmp = c0_m / math.sqrt((V * (l / A)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))));
	elseif (t_0 <= 4e+276)
		tmp = t_0;
	else
		tmp = Float64(c0_m / sqrt(Float64(V * Float64(l / A))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	elseif (t_0 <= 4e+276)
		tmp = t_0;
	else
		tmp = c0_m / sqrt((V * (l / A)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 4e+276], t$95$0, N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.4%

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

      1. associate-/r* 73.2%

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

      2. div-inv 73.2%

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

      3. div-inv 73.2%

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

      4. associate-*l* 67.6%

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

   4. Applied egg-rr 67.6%

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

      1. sqrt-prod 38.9%

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

      2. div-inv 38.9%

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

      3. associate-*r* 38.1%

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

      4. *-commutative 38.1%

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

      5. div-inv 38.1%

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

      6. frac-times 37.5%

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

      7. metadata-eval 37.5%

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

      8. sqrt-div 38.4%

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

      9. metadata-eval 38.4%

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

      10. div-inv 38.4%

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

      11. add-sqr-sqrt 6.2%

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

      12. sqrt-unprod 6.3%

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

      13. frac-times 6.3%

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

   6. Applied egg-rr 14.9%

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

      1. associate-/l* 14.9%

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

      2. associate-/r* 15.8%

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

   8. Simplified 15.8%

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

      1. associate-/l/ 14.9%

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

      2. unpow2 14.9%

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

      3. times-frac 19.5%

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

   10. Applied egg-rr 19.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e276

   1. Initial program 98.6%

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

   if 4.0000000000000002e276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.9%

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

      1. associate-/r* 76.6%

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

      2. div-inv 76.6%

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

      3. div-inv 76.6%

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

      4. associate-*l* 64.7%

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

   4. Applied egg-rr 64.7%

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

      1. sqrt-prod 33.2%

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

      2. div-inv 33.3%

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

      3. *-commutative 33.3%

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

      4. div-inv 33.2%

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

      5. frac-times 33.3%

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

      6. metadata-eval 33.3%

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

      7. sqrt-div 33.2%

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

      8. metadata-eval 33.2%

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

      9. associate-/r/ 33.2%

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

      10. un-div-inv 33.2%

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

      11. sqrt-undiv 65.9%

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

   6. Applied egg-rr 65.9%

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

      1. associate-*r/ 76.5%

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

   8. Simplified 76.5%

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

Alternative 5: 80.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-269}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+276}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (<= t_0 4e-269)
      (/ c0_m (sqrt (* l (/ V A))))
      (if (<= t_0 4e+276) t_0 (/ c0_m (sqrt (* V (/ l A)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-269) {
		tmp = c0_m / sqrt((l * (V / A)));
	} else if (t_0 <= 4e+276) {
		tmp = t_0;
	} else {
		tmp = c0_m / sqrt((V * (l / A)));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if (t_0 <= 4d-269) then
        tmp = c0_m / sqrt((l * (v / a)))
    else if (t_0 <= 4d+276) then
        tmp = t_0
    else
        tmp = c0_m / sqrt((v * (l / a)))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 4e-269) {
		tmp = c0_m / Math.sqrt((l * (V / A)));
	} else if (t_0 <= 4e+276) {
		tmp = t_0;
	} else {
		tmp = c0_m / Math.sqrt((V * (l / A)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 4e-269:
		tmp = c0_m / math.sqrt((l * (V / A)))
	elif t_0 <= 4e+276:
		tmp = t_0
	else:
		tmp = c0_m / math.sqrt((V * (l / A)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 4e-269)
		tmp = Float64(c0_m / sqrt(Float64(l * Float64(V / A))));
	elseif (t_0 <= 4e+276)
		tmp = t_0;
	else
		tmp = Float64(c0_m / sqrt(Float64(V * Float64(l / A))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 4e-269)
		tmp = c0_m / sqrt((l * (V / A)));
	elseif (t_0 <= 4e+276)
		tmp = t_0;
	else
		tmp = c0_m / sqrt((V * (l / A)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 4e-269], N[(c0$95$m / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+276], t$95$0, N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.9999999999999998e-269

   1. Initial program 67.8%

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

      1. associate-/r* 72.6%

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

      2. div-inv 72.5%

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

      3. div-inv 72.5%

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

      4. associate-*l* 68.1%

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

   4. Applied egg-rr 68.1%

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

      1. sqrt-prod 39.0%

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

      2. div-inv 39.1%

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

      3. *-commutative 39.1%

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

      4. div-inv 39.0%

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

      5. frac-times 38.4%

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

      6. metadata-eval 38.4%

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

      7. sqrt-div 39.3%

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

      8. metadata-eval 39.3%

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

      9. associate-/r/ 39.3%

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

      10. un-div-inv 39.3%

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

      11. sqrt-undiv 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. *-commutative 67.5%

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

      2. associate-/l* 71.6%

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

   8. Simplified 71.6%

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

   if 3.9999999999999998e-269 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 4.0000000000000002e276

   1. Initial program 98.5%

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

   if 4.0000000000000002e276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 65.9%

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

      1. associate-/r* 76.6%

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

      2. div-inv 76.6%

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

      3. div-inv 76.6%

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

      4. associate-*l* 64.7%

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

   4. Applied egg-rr 64.7%

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

      1. sqrt-prod 33.2%

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

      2. div-inv 33.3%

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

      3. *-commutative 33.3%

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

      4. div-inv 33.2%

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

      5. frac-times 33.3%

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

      6. metadata-eval 33.3%

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

      7. sqrt-div 33.2%

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

      8. metadata-eval 33.2%

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

      9. associate-/r/ 33.2%

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

      10. un-div-inv 33.2%

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

      11. sqrt-undiv 65.9%

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

   6. Applied egg-rr 65.9%

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

      1. associate-*r/ 76.5%

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

   8. Simplified 76.5%

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

Alternative 6: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-270}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V} \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) (- INFINITY))
    (/ c0_m (* (sqrt (/ V A)) (sqrt l)))
    (if (<= (* V l) -2e-270)
      (* c0_m (/ (sqrt (- A)) (sqrt (* l (- V)))))
      (if (<= (* V l) 4e-319)
        (* c0_m (sqrt (* (/ A V) (/ 1.0 l))))
        (if (<= (* V l) 2e+281)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (sqrt (* A (* (/ c0_m l) (/ c0_m V))))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0_m / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -2e-270) {
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	} else if ((V * l) <= 4e-319) {
		tmp = c0_m * sqrt(((A / V) * (1.0 / l)));
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0_m / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -2e-270) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((V * l) <= 4e-319) {
		tmp = c0_m * Math.sqrt(((A / V) * (1.0 / l)));
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0_m / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -2e-270:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (V * l) <= 4e-319:
		tmp = c0_m * math.sqrt(((A / V) * (1.0 / l)))
	elif (V * l) <= 2e+281:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0_m / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -2e-270)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) * Float64(1.0 / l))));
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0_m / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -2e-270)
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	elseif ((V * l) <= 4e-319)
		tmp = c0_m * sqrt(((A / V) * (1.0 / l)));
	elseif ((V * l) <= 2e+281)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0$95$m / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-270], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 36.3%

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

      1. associate-/r* 68.0%

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

      2. div-inv 68.0%

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

      3. div-inv 68.0%

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

      4. associate-*l* 36.3%

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

   4. Applied egg-rr 36.3%

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

      1. sqrt-prod 0.0%

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

      2. div-inv 0.0%

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

      3. *-commutative 0.0%

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

      4. div-inv 0.0%

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

      5. frac-times 0.0%

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

      6. metadata-eval 0.0%

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

      7. sqrt-div 0.0%

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

      8. metadata-eval 0.0%

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

      9. associate-/r/ 0.0%

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

      10. un-div-inv 0.0%

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

      11. sqrt-undiv 36.3%

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

   6. Applied egg-rr 36.3%

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

      1. *-commutative 36.3%

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

      2. associate-/l* 68.2%

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

   8. Simplified 68.2%

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

      1. *-commutative 68.2%

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

      2. sqrt-prod 52.2%

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

   10. Applied egg-rr 52.2%

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

   if -inf.0 < (*.f64 V l) < -2.0000000000000001e-270

   1. Initial program 88.7%

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

      1. frac-2neg 88.7%

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

      2. sqrt-div 99.4%

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

      3. *-commutative 99.4%

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

      4. distribute-rgt-neg-in 99.4%

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

   4. Applied egg-rr 99.4%

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

   if -2.0000000000000001e-270 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 52.8%

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

      1. associate-/r* 78.8%

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

      2. div-inv 78.8%

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

   4. Applied egg-rr 78.8%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

   if 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 19.4%

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

      1. associate-/r* 67.4%

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

      2. div-inv 67.3%

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

      3. div-inv 67.3%

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

      4. associate-*l* 25.9%

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

   4. Applied egg-rr 25.9%

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

      1. sqrt-prod 25.9%

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

      2. div-inv 25.9%

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

      3. associate-*r* 25.6%

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

      4. *-commutative 25.6%

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

      5. div-inv 25.6%

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

      6. frac-times 19.1%

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

      7. metadata-eval 19.1%

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

      8. sqrt-div 19.1%

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

      9. metadata-eval 19.1%

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

      10. div-inv 19.1%

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

      11. add-sqr-sqrt 19.1%

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

      12. sqrt-unprod 19.1%

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

      13. frac-times 18.2%

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

   6. Applied egg-rr 17.9%

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

      1. associate-/l* 18.4%

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

      2. associate-/r* 26.2%

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

   8. Simplified 26.2%

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

      1. associate-/l/ 18.4%

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

      2. unpow2 18.4%

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

      3. times-frac 46.9%

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

   10. Applied egg-rr 46.9%

       \[\leadsto \sqrt{A \cdot \color{blue}{\left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 86.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0\_m}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0\_m \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) -5e+140)
    (/ c0_m (* (sqrt (/ V A)) (sqrt l)))
    (if (<= (* V l) -2e-88)
      (/ 1.0 (/ (sqrt (/ (* V l) A)) c0_m))
      (if (<= (* V l) 4e-319)
        (* c0_m (pow (* l (/ V A)) -0.5))
        (if (<= (* V l) 2e+281)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (sqrt (* A (* (/ c0_m l) (/ c0_m V))))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+140) {
		tmp = c0_m / (sqrt((V / A)) * sqrt(l));
	} else if ((V * l) <= -2e-88) {
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0_m);
	} else if ((V * l) <= 4e-319) {
		tmp = c0_m * pow((l * (V / A)), -0.5);
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+140)) then
        tmp = c0_m / (sqrt((v / a)) * sqrt(l))
    else if ((v * l) <= (-2d-88)) then
        tmp = 1.0d0 / (sqrt(((v * l) / a)) / c0_m)
    else if ((v * l) <= 4d-319) then
        tmp = c0_m * ((l * (v / a)) ** (-0.5d0))
    else if ((v * l) <= 2d+281) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = sqrt((a * ((c0_m / l) * (c0_m / v))))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+140) {
		tmp = c0_m / (Math.sqrt((V / A)) * Math.sqrt(l));
	} else if ((V * l) <= -2e-88) {
		tmp = 1.0 / (Math.sqrt(((V * l) / A)) / c0_m);
	} else if ((V * l) <= 4e-319) {
		tmp = c0_m * Math.pow((l * (V / A)), -0.5);
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -5e+140:
		tmp = c0_m / (math.sqrt((V / A)) * math.sqrt(l))
	elif (V * l) <= -2e-88:
		tmp = 1.0 / (math.sqrt(((V * l) / A)) / c0_m)
	elif (V * l) <= 4e-319:
		tmp = c0_m * math.pow((l * (V / A)), -0.5)
	elif (V * l) <= 2e+281:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+140)
		tmp = Float64(c0_m / Float64(sqrt(Float64(V / A)) * sqrt(l)));
	elseif (Float64(V * l) <= -2e-88)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(V * l) / A)) / c0_m));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(c0_m * (Float64(l * Float64(V / A)) ^ -0.5));
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+140)
		tmp = c0_m / (sqrt((V / A)) * sqrt(l));
	elseif ((V * l) <= -2e-88)
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0_m);
	elseif ((V * l) <= 4e-319)
		tmp = c0_m * ((l * (V / A)) ^ -0.5);
	elseif ((V * l) <= 2e+281)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -5e+140], N[(c0$95$m / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-88], N[(1.0 / N[(N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] / c0$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(c0$95$m * N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.00000000000000008e140

   1. Initial program 63.6%

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

      1. associate-/r* 73.0%

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

      2. div-inv 73.0%

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

      3. div-inv 73.0%

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

      4. associate-*l* 63.5%

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

   4. Applied egg-rr 63.5%

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

      1. sqrt-prod 0.0%

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

      2. div-inv 0.0%

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

      3. *-commutative 0.0%

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

      4. div-inv 0.0%

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

      5. frac-times 0.0%

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

      6. metadata-eval 0.0%

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

      7. sqrt-div 0.0%

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

      8. metadata-eval 0.0%

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

      9. associate-/r/ 0.0%

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

      10. un-div-inv 0.0%

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

      11. sqrt-undiv 61.9%

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

   6. Applied egg-rr 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-/l* 72.1%

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

   8. Simplified 72.1%

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

      1. *-commutative 72.1%

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

      2. sqrt-prod 52.6%

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

   10. Applied egg-rr 52.6%

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

   if -5.00000000000000008e140 < (*.f64 V l) < -1.99999999999999987e-88

   1. Initial program 99.5%

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

      1. associate-/r* 84.9%

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

      2. div-inv 84.9%

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

      3. div-inv 84.9%

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

      4. associate-*l* 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. sqrt-prod 0.0%

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

      2. div-inv 0.0%

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

      3. associate-*r* 0.0%

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

      4. *-commutative 0.0%

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

      5. div-inv 0.0%

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

      6. frac-times 0.0%

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

      7. metadata-eval 0.0%

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

      8. sqrt-div 0.0%

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

      9. metadata-eval 0.0%

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

      10. div-inv 0.0%

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

      11. clear-num 0.0%

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

      12. associate-/r* 0.0%

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

      13. sqrt-undiv 99.6%

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

   6. Applied egg-rr 99.6%

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

   if -1.99999999999999987e-88 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 66.7%

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

      1. associate-/r* 75.4%

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

      2. div-inv 75.4%

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

      3. div-inv 75.4%

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

      4. associate-*l* 65.2%

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

   4. Applied egg-rr 65.2%

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

      1. sqrt-prod 8.9%

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

      2. div-inv 8.9%

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

      3. *-commutative 8.9%

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

      4. div-inv 8.9%

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

      5. frac-times 8.9%

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

      6. metadata-eval 8.9%

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

      7. sqrt-div 9.8%

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

      8. metadata-eval 9.8%

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

      9. associate-/r/ 9.8%

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

      10. un-div-inv 9.8%

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

      11. sqrt-undiv 66.7%

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

   6. Applied egg-rr 66.7%

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

      1. associate-*r/ 77.4%

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

   8. Simplified 77.4%

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

      1. clear-num 77.4%

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

      2. associate-*r/ 66.7%

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

      3. associate-/r/ 66.7%

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

      4. pow1/2 66.7%

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

      5. associate-*r/ 77.4%

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

      6. pow-flip 77.5%

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

      7. associate-*r/ 66.8%

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

      8. *-commutative 66.8%

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

      9. *-un-lft-identity 66.8%

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

      10. times-frac 75.5%

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

      11. /-rgt-identity 75.5%

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

      12. metadata-eval 75.5%

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

   10. Applied egg-rr 75.5%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

   if 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 19.4%

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

      1. associate-/r* 67.4%

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

      2. div-inv 67.3%

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

      3. div-inv 67.3%

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

      4. associate-*l* 25.9%

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

   4. Applied egg-rr 25.9%

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

      1. sqrt-prod 25.9%

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

      2. div-inv 25.9%

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

      3. associate-*r* 25.6%

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

      4. *-commutative 25.6%

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

      5. div-inv 25.6%

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

      6. frac-times 19.1%

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

      7. metadata-eval 19.1%

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

      8. sqrt-div 19.1%

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

      9. metadata-eval 19.1%

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

      10. div-inv 19.1%

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

      11. add-sqr-sqrt 19.1%

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

      12. sqrt-unprod 19.1%

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

      13. frac-times 18.2%

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

   6. Applied egg-rr 17.9%

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

      1. associate-/l* 18.4%

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

      2. associate-/r* 26.2%

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

   8. Simplified 26.2%

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

      1. associate-/l/ 18.4%

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

      2. unpow2 18.4%

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

      3. times-frac 46.9%

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

   10. Applied egg-rr 46.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+140}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{A}} \cdot \sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+140}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0\_m}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0\_m \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) -5e+140)
    (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
    (if (<= (* V l) -2e-88)
      (/ 1.0 (/ (sqrt (/ (* V l) A)) c0_m))
      (if (<= (* V l) 4e-319)
        (* c0_m (pow (* l (/ V A)) -0.5))
        (if (<= (* V l) 2e+281)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (sqrt (* A (* (/ c0_m l) (/ c0_m V))))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+140) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -2e-88) {
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0_m);
	} else if ((V * l) <= 4e-319) {
		tmp = c0_m * pow((l * (V / A)), -0.5);
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-5d+140)) then
        tmp = c0_m * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-2d-88)) then
        tmp = 1.0d0 / (sqrt(((v * l) / a)) / c0_m)
    else if ((v * l) <= 4d-319) then
        tmp = c0_m * ((l * (v / a)) ** (-0.5d0))
    else if ((v * l) <= 2d+281) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = sqrt((a * ((c0_m / l) * (c0_m / v))))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -5e+140) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -2e-88) {
		tmp = 1.0 / (Math.sqrt(((V * l) / A)) / c0_m);
	} else if ((V * l) <= 4e-319) {
		tmp = c0_m * Math.pow((l * (V / A)), -0.5);
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -5e+140:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -2e-88:
		tmp = 1.0 / (math.sqrt(((V * l) / A)) / c0_m)
	elif (V * l) <= 4e-319:
		tmp = c0_m * math.pow((l * (V / A)), -0.5)
	elif (V * l) <= 2e+281:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -5e+140)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -2e-88)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(V * l) / A)) / c0_m));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(c0_m * (Float64(l * Float64(V / A)) ^ -0.5));
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -5e+140)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -2e-88)
		tmp = 1.0 / (sqrt(((V * l) / A)) / c0_m);
	elseif ((V * l) <= 4e-319)
		tmp = c0_m * ((l * (V / A)) ^ -0.5);
	elseif ((V * l) <= 2e+281)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = sqrt((A * ((c0_m / l) * (c0_m / V))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -5e+140], N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-88], N[(1.0 / N[(N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision] / c0$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(c0$95$m * N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if (*.f64 V l) < -5.00000000000000008e140

   1. Initial program 63.6%

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

      1. associate-/r* 73.0%

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

      2. sqrt-div 52.6%

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

      3. associate-*r/ 49.2%

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

   4. Applied egg-rr 49.2%

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

      1. associate-/l* 52.6%

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

   6. Simplified 52.6%

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

   if -5.00000000000000008e140 < (*.f64 V l) < -1.99999999999999987e-88

   1. Initial program 99.5%

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

      1. associate-/r* 84.9%

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

      2. div-inv 84.9%

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

      3. div-inv 84.9%

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

      4. associate-*l* 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. sqrt-prod 0.0%

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

      2. div-inv 0.0%

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

      3. associate-*r* 0.0%

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

      4. *-commutative 0.0%

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

      5. div-inv 0.0%

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

      6. frac-times 0.0%

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

      7. metadata-eval 0.0%

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

      8. sqrt-div 0.0%

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

      9. metadata-eval 0.0%

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

      10. div-inv 0.0%

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

      11. clear-num 0.0%

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

      12. associate-/r* 0.0%

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

      13. sqrt-undiv 99.6%

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

   6. Applied egg-rr 99.6%

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

   if -1.99999999999999987e-88 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 66.7%

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

      1. associate-/r* 75.4%

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

      2. div-inv 75.4%

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

      3. div-inv 75.4%

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

      4. associate-*l* 65.2%

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

   4. Applied egg-rr 65.2%

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

      1. sqrt-prod 8.9%

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

      2. div-inv 8.9%

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

      3. *-commutative 8.9%

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

      4. div-inv 8.9%

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

      5. frac-times 8.9%

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

      6. metadata-eval 8.9%

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

      7. sqrt-div 9.8%

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

      8. metadata-eval 9.8%

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

      9. associate-/r/ 9.8%

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

      10. un-div-inv 9.8%

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

      11. sqrt-undiv 66.7%

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

   6. Applied egg-rr 66.7%

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

      1. associate-*r/ 77.4%

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

   8. Simplified 77.4%

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

      1. clear-num 77.4%

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

      2. associate-*r/ 66.7%

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

      3. associate-/r/ 66.7%

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

      4. pow1/2 66.7%

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

      5. associate-*r/ 77.4%

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

      6. pow-flip 77.5%

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

      7. associate-*r/ 66.8%

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

      8. *-commutative 66.8%

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

      9. *-un-lft-identity 66.8%

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

      10. times-frac 75.5%

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

      11. /-rgt-identity 75.5%

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

      12. metadata-eval 75.5%

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

   10. Applied egg-rr 75.5%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

   if 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 19.4%

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

      1. associate-/r* 67.4%

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

      2. div-inv 67.3%

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

      3. div-inv 67.3%

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

      4. associate-*l* 25.9%

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

   4. Applied egg-rr 25.9%

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

      1. sqrt-prod 25.9%

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

      2. div-inv 25.9%

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

      3. associate-*r* 25.6%

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

      4. *-commutative 25.6%

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

      5. div-inv 25.6%

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

      6. frac-times 19.1%

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

      7. metadata-eval 19.1%

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

      8. sqrt-div 19.1%

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

      9. metadata-eval 19.1%

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

      10. div-inv 19.1%

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

      11. add-sqr-sqrt 19.1%

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

      12. sqrt-unprod 19.1%

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

      13. frac-times 18.2%

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

   6. Applied egg-rr 17.9%

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

      1. associate-/l* 18.4%

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

      2. associate-/r* 26.2%

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

   8. Simplified 26.2%

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

      1. associate-/l/ 18.4%

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

      2. unpow2 18.4%

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

      3. times-frac 46.9%

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

   10. Applied egg-rr 46.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+140}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{\frac{V \cdot \ell}{A}}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{A \cdot \left(\frac{c0\_m}{\ell} \cdot \frac{c0\_m}{V}\right)}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+276}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-88}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0\_m \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (sqrt (* A (* (/ c0_m l) (/ c0_m V))))))
   (*
    c0_s
    (if (<= (* V l) -5e+276)
      t_0
      (if (<= (* V l) -2e-88)
        (* c0_m (sqrt (/ A (* V l))))
        (if (<= (* V l) 4e-319)
          (* c0_m (pow (* l (/ V A)) -0.5))
          (if (<= (* V l) 2e+281)
            (* c0_m (/ (sqrt A) (sqrt (* V l))))
            t_0)))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = sqrt((A * ((c0_m / l) * (c0_m / V))));
	double tmp;
	if ((V * l) <= -5e+276) {
		tmp = t_0;
	} else if ((V * l) <= -2e-88) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 4e-319) {
		tmp = c0_m * pow((l * (V / A)), -0.5);
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a * ((c0_m / l) * (c0_m / v))))
    if ((v * l) <= (-5d+276)) then
        tmp = t_0
    else if ((v * l) <= (-2d-88)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 4d-319) then
        tmp = c0_m * ((l * (v / a)) ** (-0.5d0))
    else if ((v * l) <= 2d+281) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = Math.sqrt((A * ((c0_m / l) * (c0_m / V))));
	double tmp;
	if ((V * l) <= -5e+276) {
		tmp = t_0;
	} else if ((V * l) <= -2e-88) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 4e-319) {
		tmp = c0_m * Math.pow((l * (V / A)), -0.5);
	} else if ((V * l) <= 2e+281) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = math.sqrt((A * ((c0_m / l) * (c0_m / V))))
	tmp = 0
	if (V * l) <= -5e+276:
		tmp = t_0
	elif (V * l) <= -2e-88:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 4e-319:
		tmp = c0_m * math.pow((l * (V / A)), -0.5)
	elif (V * l) <= 2e+281:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = t_0
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = sqrt(Float64(A * Float64(Float64(c0_m / l) * Float64(c0_m / V))))
	tmp = 0.0
	if (Float64(V * l) <= -5e+276)
		tmp = t_0;
	elseif (Float64(V * l) <= -2e-88)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 4e-319)
		tmp = Float64(c0_m * (Float64(l * Float64(V / A)) ^ -0.5));
	elseif (Float64(V * l) <= 2e+281)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = sqrt((A * ((c0_m / l) * (c0_m / V))));
	tmp = 0.0;
	if ((V * l) <= -5e+276)
		tmp = t_0;
	elseif ((V * l) <= -2e-88)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 4e-319)
		tmp = c0_m * ((l * (V / A)) ^ -0.5);
	elseif ((V * l) <= 2e+281)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	else
		tmp = t_0;
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[N[(A * N[(N[(c0$95$m / l), $MachinePrecision] * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -5e+276], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -2e-88], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 4e-319], N[(c0$95$m * N[Power[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+281], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000001e276 or 2.0000000000000001e281 < (*.f64 V l)

   1. Initial program 37.9%

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

      1. associate-/r* 67.6%

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

      2. div-inv 67.6%

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

      3. div-inv 67.6%

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

      4. associate-*l* 40.0%

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

   4. Applied egg-rr 40.0%

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

      1. sqrt-prod 8.1%

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

      2. div-inv 8.1%

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

      3. associate-*r* 8.0%

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

      4. *-commutative 8.0%

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

      5. div-inv 8.0%

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

      6. frac-times 5.9%

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

      7. metadata-eval 5.9%

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

      8. sqrt-div 5.9%

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

      9. metadata-eval 5.9%

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

      10. div-inv 5.9%

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

      11. add-sqr-sqrt 5.9%

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

      12. sqrt-unprod 5.9%

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

      13. frac-times 5.7%

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

   6. Applied egg-rr 29.3%

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

      1. associate-/l* 31.5%

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

      2. associate-/r* 34.0%

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

   8. Simplified 34.0%

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

      1. associate-/l/ 31.5%

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

      2. unpow2 31.5%

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

      3. times-frac 46.9%

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

   10. Applied egg-rr 46.9%

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

   if -5.00000000000000001e276 < (*.f64 V l) < -1.99999999999999987e-88

   1. Initial program 95.5%

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

   if -1.99999999999999987e-88 < (*.f64 V l) < 4.0000049e-319

   1. Initial program 66.7%

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

      1. associate-/r* 75.4%

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

      2. div-inv 75.4%

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

      3. div-inv 75.4%

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

      4. associate-*l* 65.2%

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

   4. Applied egg-rr 65.2%

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

      1. sqrt-prod 8.9%

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

      2. div-inv 8.9%

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

      3. *-commutative 8.9%

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

      4. div-inv 8.9%

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

      5. frac-times 8.9%

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

      6. metadata-eval 8.9%

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

      7. sqrt-div 9.8%

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

      8. metadata-eval 9.8%

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

      9. associate-/r/ 9.8%

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

      10. un-div-inv 9.8%

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

      11. sqrt-undiv 66.7%

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

   6. Applied egg-rr 66.7%

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

      1. associate-*r/ 77.4%

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

   8. Simplified 77.4%

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

      1. clear-num 77.4%

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

      2. associate-*r/ 66.7%

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

      3. associate-/r/ 66.7%

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

      4. pow1/2 66.7%

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

      5. associate-*r/ 77.4%

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

      6. pow-flip 77.5%

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

      7. associate-*r/ 66.8%

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

      8. *-commutative 66.8%

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

      9. *-un-lft-identity 66.8%

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

      10. times-frac 75.5%

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

      11. /-rgt-identity 75.5%

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

      12. metadata-eval 75.5%

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

   10. Applied egg-rr 75.5%

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

   if 4.0000049e-319 < (*.f64 V l) < 2.0000000000000001e281

   1. Initial program 87.4%

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

      1. sqrt-div 99.5%

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

      2. associate-*r/ 98.1%

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

   4. Applied egg-rr 98.1%

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

      1. associate-/l* 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-88}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 4 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{-0.5}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+281}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot \left(\frac{c0}{\ell} \cdot \frac{c0}{V}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \left(c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	return Float64(c0_s * Float64(c0_m * sqrt(Float64(A / Float64(V * l)))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))