Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.5% → 92.7%

Time: 10.7s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{-A}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{t\_0}{\sqrt{-V}} \cdot \frac{c0\_m}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0\_m \cdot \frac{t\_0}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+301}\right):\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (sqrt (- A))))
   (*
    c0_s
    (if (<= (* V l) (- INFINITY))
      (* (/ t_0 (sqrt (- V))) (/ c0_m (sqrt l)))
      (if (<= (* V l) -1e-304)
        (* c0_m (/ t_0 (sqrt (* V (- l)))))
        (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+301)))
          (sqrt (/ (* c0_m (/ A l)) (/ V c0_m)))
          (/ c0_m (/ (sqrt (* V l)) (sqrt A)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = sqrt(-A);
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = (t_0 / sqrt(-V)) * (c0_m / sqrt(l));
	} else if ((V * l) <= -1e-304) {
		tmp = c0_m * (t_0 / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+301)) {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} else {
		tmp = c0_m / (sqrt((V * l)) / sqrt(A));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = Math.sqrt(-A);
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = (t_0 / Math.sqrt(-V)) * (c0_m / Math.sqrt(l));
	} else if ((V * l) <= -1e-304) {
		tmp = c0_m * (t_0 / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+301)) {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} else {
		tmp = c0_m / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = math.sqrt(-A)
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = (t_0 / math.sqrt(-V)) * (c0_m / math.sqrt(l))
	elif (V * l) <= -1e-304:
		tmp = c0_m * (t_0 / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+301):
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	else:
		tmp = c0_m / (math.sqrt((V * l)) / math.sqrt(A))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = sqrt(Float64(-A))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(Float64(t_0 / sqrt(Float64(-V))) * Float64(c0_m / sqrt(l)));
	elseif (Float64(V * l) <= -1e-304)
		tmp = Float64(c0_m * Float64(t_0 / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+301))
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	else
		tmp = Float64(c0_m / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = sqrt(-A);
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (t_0 / sqrt(-V)) * (c0_m / sqrt(l));
	elseif ((V * l) <= -1e-304)
		tmp = c0_m * (t_0 / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+301)))
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	else
		tmp = c0_m / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[Sqrt[(-A)], $MachinePrecision]}, N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[(t$95$0 / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-304], N[(c0$95$m * N[(t$95$0 / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+301]], $MachinePrecision]], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 48.2%

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

      1. associate-/r* 77.9%

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

      2. sqrt-div 38.3%

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

      3. associate-*r/ 38.5%

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

   4. Applied egg-rr 38.5%

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

      1. *-commutative 38.5%

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

      2. associate-/l* 38.5%

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

   6. Simplified 38.5%

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

      1. frac-2neg 38.5%

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

      2. sqrt-div 38.5%

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

   8. Applied egg-rr 38.5%

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

   if -inf.0 < (*.f64 V l) < -9.99999999999999971e-305

   1. Initial program 80.9%

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

      1. frac-2neg 80.9%

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

      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 -9.99999999999999971e-305 < (*.f64 V l) < 0.0 or 1.00000000000000005e301 < (*.f64 V l)

   1. Initial program 34.5%

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

      1. add-sqr-sqrt 24.7%

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

      2. sqrt-unprod 24.8%

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

      3. *-commutative 24.8%

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

      4. *-commutative 24.8%

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

      5. swap-sqr 24.4%

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

      6. add-sqr-sqrt 24.4%

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

      7. pow2 24.4%

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

   4. Applied egg-rr 24.4%

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

      1. associate-*l/ 22.3%

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

      2. *-commutative 22.3%

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

      3. times-frac 37.6%

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

   6. Simplified 37.6%

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

      1. unpow2 37.6%

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

      2. *-un-lft-identity 37.6%

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

      3. times-frac 41.8%

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

   8. Applied egg-rr 41.8%

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

      1. /-rgt-identity 41.8%

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

      2. associate-*r* 43.8%

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

      3. clear-num 43.8%

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

      4. un-div-inv 43.8%

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

   10. Applied egg-rr 43.8%

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

   if 0.0 < (*.f64 V l) < 1.00000000000000005e301

   1. Initial program 85.4%

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

      1. associate-/r* 75.8%

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

      2. div-inv 75.7%

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

      3. div-inv 75.7%

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

      4. associate-*l* 85.4%

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

   4. Applied egg-rr 85.4%

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

      1. associate-*l/ 85.4%

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

      2. *-un-lft-identity 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-*r/ 71.8%

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

      2. div-inv 71.8%

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

      3. clear-num 70.6%

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

      4. metadata-eval 70.6%

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

      5. un-div-inv 70.6%

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

      6. clear-num 71.4%

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

      7. add-sqr-sqrt 71.2%

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

      8. frac-times 71.3%

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

      9. sqrt-unprod 71.1%

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

      10. add-sqr-sqrt 71.3%

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

      11. un-div-inv 71.4%

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

   8. Applied egg-rr 71.4%

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

      1. associate-*r/ 83.4%

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

      2. sqrt-div 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{-A}}{\sqrt{-V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+301}\right):\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+301}\right):\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) (- INFINITY))
    (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
    (if (<= (* V l) -1e-304)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+301)))
        (sqrt (/ (* c0_m (/ A l)) (/ V c0_m)))
        (/ c0_m (/ (sqrt (* V l)) (sqrt A))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -1e-304) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+301)) {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} else {
		tmp = c0_m / (sqrt((V * l)) / sqrt(A));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -1e-304) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+301)) {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} else {
		tmp = c0_m / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -1e-304:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+301):
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	else:
		tmp = c0_m / (math.sqrt((V * l)) / math.sqrt(A))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -1e-304)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+301))
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	else
		tmp = Float64(c0_m / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -1e-304)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+301)))
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	else
		tmp = c0_m / (sqrt((V * l)) / sqrt(A));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-304], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+301]], $MachinePrecision]], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 48.2%

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

      1. associate-/r* 77.9%

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

      2. sqrt-div 38.3%

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

      3. associate-*r/ 38.5%

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

   4. Applied egg-rr 38.5%

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

      1. associate-/l* 38.3%

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

   6. Simplified 38.3%

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

   if -inf.0 < (*.f64 V l) < -9.99999999999999971e-305

   1. Initial program 80.9%

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

      1. frac-2neg 80.9%

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

      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 -9.99999999999999971e-305 < (*.f64 V l) < 0.0 or 1.00000000000000005e301 < (*.f64 V l)

   1. Initial program 34.5%

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

      1. add-sqr-sqrt 24.7%

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

      2. sqrt-unprod 24.8%

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

      3. *-commutative 24.8%

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

      4. *-commutative 24.8%

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

      5. swap-sqr 24.4%

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

      6. add-sqr-sqrt 24.4%

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

      7. pow2 24.4%

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

   4. Applied egg-rr 24.4%

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

      1. associate-*l/ 22.3%

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

      2. *-commutative 22.3%

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

      3. times-frac 37.6%

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

   6. Simplified 37.6%

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

      1. unpow2 37.6%

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

      2. *-un-lft-identity 37.6%

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

      3. times-frac 41.8%

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

   8. Applied egg-rr 41.8%

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

      1. /-rgt-identity 41.8%

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

      2. associate-*r* 43.8%

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

      3. clear-num 43.8%

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

      4. un-div-inv 43.8%

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

   10. Applied egg-rr 43.8%

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

   if 0.0 < (*.f64 V l) < 1.00000000000000005e301

   1. Initial program 85.4%

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

      1. associate-/r* 75.8%

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

      2. div-inv 75.7%

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

      3. div-inv 75.7%

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

      4. associate-*l* 85.4%

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

   4. Applied egg-rr 85.4%

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

      1. associate-*l/ 85.4%

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

      2. *-un-lft-identity 85.4%

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

   6. Applied egg-rr 85.4%

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

      1. associate-*r/ 71.8%

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

      2. div-inv 71.8%

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

      3. clear-num 70.6%

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

      4. metadata-eval 70.6%

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

      5. un-div-inv 70.6%

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

      6. clear-num 71.4%

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

      7. add-sqr-sqrt 71.2%

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

      8. frac-times 71.3%

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

      9. sqrt-unprod 71.1%

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

      10. add-sqr-sqrt 71.3%

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

      11. un-div-inv 71.4%

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

   8. Applied egg-rr 71.4%

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

      1. associate-*r/ 83.4%

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

      2. sqrt-div 98.7%

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

   10. Applied egg-rr 98.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+301}\right):\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-310}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 1e-310)
      (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
      (if (<= t_0 5e+306)
        (* c0_m (sqrt t_0))
        (sqrt (/ (* c0_m (/ A l)) (/ V c0_m))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-310) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 1d-310) then
        tmp = c0_m * (sqrt((a / v)) / sqrt(l))
    else if (t_0 <= 5d+306) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = sqrt(((c0_m * (a / l)) / (v / c0_m)))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-310) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 1e-310:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif t_0 <= 5e+306:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 1e-310)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (t_0 <= 5e+306)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-310)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif (t_0 <= 5e+306)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 1e-310], N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.8%

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

      1. associate-/r* 50.0%

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

      2. sqrt-div 34.4%

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

      3. associate-*r/ 34.3%

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

   4. Applied egg-rr 34.3%

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

      1. associate-/l* 34.4%

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

   6. Simplified 34.4%

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

   if 9.999999999999969e-311 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 99.5%

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

   if 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 36.9%

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

      1. add-sqr-sqrt 18.9%

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

      2. sqrt-unprod 19.1%

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

      3. *-commutative 19.1%

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

      4. *-commutative 19.1%

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

      5. swap-sqr 18.8%

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

      6. add-sqr-sqrt 18.8%

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

      7. pow2 18.8%

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

   4. Applied egg-rr 18.8%

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

      1. associate-*l/ 21.3%

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

      2. *-commutative 21.3%

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

      3. times-frac 27.2%

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

   6. Simplified 27.2%

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

      1. unpow2 27.2%

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

      2. *-un-lft-identity 27.2%

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

      3. times-frac 29.0%

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

   8. Applied egg-rr 29.0%

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

      1. /-rgt-identity 29.0%

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

      2. associate-*r* 30.9%

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

      3. clear-num 30.9%

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

      4. un-div-inv 30.9%

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

   10. Applied egg-rr 30.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;\sqrt{\frac{c0\_m \cdot \frac{A}{\ell}}{\frac{V}{c0\_m}}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 5e+306)))
      (sqrt (/ (* c0_m (/ A l)) (/ V c0_m)))
      (* c0_m (sqrt t_0))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+306))) then
        tmp = sqrt(((c0_m * (a / l)) / (v / c0_m)))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = Math.sqrt(((c0_m * (A / l)) / (V / c0_m)));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+306):
		tmp = math.sqrt(((c0_m * (A / l)) / (V / c0_m)))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306))
		tmp = sqrt(Float64(Float64(c0_m * Float64(A / l)) / Float64(V / c0_m)));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+306)))
		tmp = sqrt(((c0_m * (A / l)) / (V / c0_m)));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+306]], $MachinePrecision]], N[Sqrt[N[(N[(c0$95$m * N[(A / l), $MachinePrecision]), $MachinePrecision] / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 35.6%

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

      1. add-sqr-sqrt 27.1%

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

      2. sqrt-unprod 27.2%

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

      3. *-commutative 27.2%

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

      4. *-commutative 27.2%

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

      5. swap-sqr 26.7%

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

      6. add-sqr-sqrt 26.7%

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

      7. pow2 26.7%

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

   4. Applied egg-rr 26.7%

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

      1. associate-*l/ 30.6%

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

      2. *-commutative 30.6%

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

      3. times-frac 34.5%

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

   6. Simplified 34.5%

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

      1. unpow2 34.5%

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

      2. *-un-lft-identity 34.5%

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

      3. times-frac 40.0%

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

   8. Applied egg-rr 40.0%

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

      1. /-rgt-identity 40.0%

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

      2. associate-*r* 41.8%

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

      3. clear-num 41.8%

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

      4. un-div-inv 41.8%

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

   10. Applied egg-rr 41.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 98.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;\sqrt{\frac{c0 \cdot \frac{A}{\ell}}{\frac{V}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \frac{c0\_m}{\frac{V}{c0\_m}}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 5e+306)))
      (sqrt (* (/ A l) (/ c0_m (/ V c0_m))))
      (* c0_m (sqrt t_0))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = sqrt(((A / l) * (c0_m / (V / c0_m))));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+306))) then
        tmp = sqrt(((a / l) * (c0_m / (v / c0_m))))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = Math.sqrt(((A / l) * (c0_m / (V / c0_m))));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+306):
		tmp = math.sqrt(((A / l) * (c0_m / (V / c0_m))))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m / Float64(V / c0_m))));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+306)))
		tmp = sqrt(((A / l) * (c0_m / (V / c0_m))));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+306]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m / N[(V / c0$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 35.6%

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

      1. add-sqr-sqrt 27.1%

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

      2. sqrt-unprod 27.2%

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

      3. *-commutative 27.2%

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

      4. *-commutative 27.2%

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

      5. swap-sqr 26.7%

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

      6. add-sqr-sqrt 26.7%

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

      7. pow2 26.7%

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

   4. Applied egg-rr 26.7%

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

      1. associate-*l/ 30.6%

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

      2. *-commutative 30.6%

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

      3. times-frac 34.5%

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

   6. Simplified 34.5%

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

      1. unpow2 34.5%

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

      2. *-un-lft-identity 34.5%

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

      3. times-frac 40.0%

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

   8. Applied egg-rr 40.0%

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

      1. /-rgt-identity 40.0%

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

      2. clear-num 40.0%

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

      3. un-div-inv 40.0%

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

   10. Applied egg-rr 40.0%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 98.7%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \frac{c0}{\frac{V}{c0}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 0.0)
      (/ c0_m (sqrt (/ l (/ A V))))
      (if (<= t_0 5e+306)
        (* c0_m (sqrt t_0))
        (* c0_m (pow (* V (/ l A)) -0.5)))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m / sqrt((l / (A / V)));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = c0_m * pow((V * (l / A)), -0.5);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0_m / sqrt((l / (a / v)))
    else if (t_0 <= 5d+306) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = c0_m * ((v * (l / a)) ** (-0.5d0))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = c0_m * Math.pow((V * (l / A)), -0.5);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0_m / math.sqrt((l / (A / V)))
	elif t_0 <= 5e+306:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = c0_m * math.pow((V * (l / A)), -0.5)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	elseif (t_0 <= 5e+306)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = Float64(c0_m * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0_m / sqrt((l / (A / V)));
	elseif (t_0 <= 5e+306)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = c0_m * ((V * (l / A)) ^ -0.5);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 34.5%

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

      1. associate-/r* 49.8%

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

      2. div-inv 49.8%

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

      3. div-inv 49.8%

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

      4. associate-*l* 38.2%

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

   4. Applied egg-rr 38.2%

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

      1. div-inv 38.2%

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

      2. *-commutative 38.2%

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

      3. associate-*l/ 49.8%

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

      4. frac-2neg 49.8%

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

      5. sqrt-div 35.8%

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

      6. associate-*l/ 35.8%

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

      7. *-un-lft-identity 35.8%

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

      8. distribute-neg-frac2 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. distribute-frac-neg2 35.8%

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

      2. distribute-frac-neg 35.8%

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

   8. Simplified 35.8%

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

      1. clear-num 35.8%

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

      2. un-div-inv 35.8%

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

      3. sqrt-undiv 49.9%

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

      4. add-sqr-sqrt 23.9%

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

      5. sqrt-unprod 17.1%

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

      6. sqr-neg 17.1%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 9.6%

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

      11. sqr-neg 9.6%

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

      12. sqrt-unprod 16.5%

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

      13. add-sqr-sqrt 49.9%

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

   10. Applied egg-rr 49.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 98.7%

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

   if 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 36.9%

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

      1. associate-/r* 48.4%

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

      2. div-inv 48.4%

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

      3. div-inv 48.4%

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

      4. associate-*l* 36.9%

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

   4. Applied egg-rr 36.9%

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

      1. associate-*l/ 36.9%

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

      2. *-un-lft-identity 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. associate-*r/ 48.3%

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

      2. div-inv 48.4%

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

      3. clear-num 48.4%

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

      4. metadata-eval 48.4%

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

      5. un-div-inv 48.4%

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

      6. clear-num 48.4%

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

      7. add-sqr-sqrt 48.4%

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

      8. frac-times 48.4%

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

      9. sqrt-unprod 50.2%

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

      10. add-sqr-sqrt 50.3%

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

      11. inv-pow 50.3%

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

      12. sqrt-pow2 50.3%

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

      13. metadata-eval 50.3%

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

   8. Applied egg-rr 50.3%

      \[\leadsto c0 \cdot \color{blue}{{\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 5e+306)))
      (* c0_m (sqrt (/ (/ A l) V)))
      (* c0_m (sqrt t_0))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+306))) then
        tmp = c0_m * sqrt(((a / l) / v))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306)) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+306):
		tmp = c0_m * math.sqrt(((A / l) / V))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+306))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+306)))
		tmp = c0_m * sqrt(((A / l) / V));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+306]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 35.6%

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

   3. Taylor expanded in c0 around 0 35.6%

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

      1. *-commutative 35.6%

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

      2. associate-/r* 49.1%

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

   5. Simplified 49.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 98.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-310} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 1e-310) (not (<= t_0 5e+300)))
      (* c0_m (sqrt (/ (/ A V) l)))
      (* c0_m (sqrt t_0))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 1e-310) || !(t_0 <= 5e+300)) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 1d-310) .or. (.not. (t_0 <= 5d+300))) then
        tmp = c0_m * sqrt(((a / v) / l))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 1e-310) || !(t_0 <= 5e+300)) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 1e-310) or not (t_0 <= 5e+300):
		tmp = c0_m * math.sqrt(((A / V) / l))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 1e-310) || !(t_0 <= 5e+300))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 1e-310) || ~((t_0 <= 5e+300)))
		tmp = c0_m * sqrt(((A / V) / l));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 1e-310], N[Not[LessEqual[t$95$0, 5e+300]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 9.999999999999969e-311 or 5.00000000000000026e300 < (/.f64 A (*.f64 V l))

   1. Initial program 36.9%

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

      1. associate-/r* 48.8%

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

   3. Simplified 48.8%

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

   if 9.999999999999969e-311 < (/.f64 A (*.f64 V l)) < 5.00000000000000026e300

   1. Initial program 99.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-310} \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{\ell}{\frac{A}{V}}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

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 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 0.0)
      (/ c0_m (sqrt (/ l (/ A V))))
      (if (<= t_0 5e+306)
        (* c0_m (sqrt 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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m / sqrt((l / (A / V)));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * sqrt(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 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0_m / sqrt((l / (a / v)))
    else if (t_0 <= 5d+306) then
        tmp = c0_m * sqrt(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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m / Math.sqrt((l / (A / V)));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * Math.sqrt(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 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0_m / math.sqrt((l / (A / V)))
	elif t_0 <= 5e+306:
		tmp = c0_m * math.sqrt(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(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0_m / sqrt(Float64(l / Float64(A / V))));
	elseif (t_0 <= 5e+306)
		tmp = Float64(c0_m * sqrt(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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0_m / sqrt((l / (A / V)));
	elseif (t_0 <= 5e+306)
		tmp = c0_m * sqrt(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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[(c0$95$m / N[Sqrt[N[(l / N[(A / V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 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 A (*.f64 V l)) < 0.0

   1. Initial program 34.5%

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

      1. associate-/r* 49.8%

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

      2. div-inv 49.8%

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

      3. div-inv 49.8%

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

      4. associate-*l* 38.2%

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

   4. Applied egg-rr 38.2%

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

      1. div-inv 38.2%

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

      2. *-commutative 38.2%

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

      3. associate-*l/ 49.8%

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

      4. frac-2neg 49.8%

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

      5. sqrt-div 35.8%

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

      6. associate-*l/ 35.8%

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

      7. *-un-lft-identity 35.8%

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

      8. distribute-neg-frac2 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. distribute-frac-neg2 35.8%

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

      2. distribute-frac-neg 35.8%

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

   8. Simplified 35.8%

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

      1. clear-num 35.8%

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

      2. un-div-inv 35.8%

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

      3. sqrt-undiv 49.9%

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

      4. add-sqr-sqrt 23.9%

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

      5. sqrt-unprod 17.1%

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

      6. sqr-neg 17.1%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 9.6%

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

      11. sqr-neg 9.6%

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

      12. sqrt-unprod 16.5%

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

      13. add-sqr-sqrt 49.9%

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

   10. Applied egg-rr 49.9%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 98.7%

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

   if 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 36.9%

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

      1. associate-/r* 48.4%

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

      2. div-inv 48.4%

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

      3. div-inv 48.4%

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

      4. associate-*l* 36.9%

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

   4. Applied egg-rr 36.9%

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

      1. associate-*l/ 36.9%

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

      2. *-un-lft-identity 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. associate-*r/ 48.3%

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

      2. div-inv 48.4%

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

      3. clear-num 48.4%

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

      4. metadata-eval 48.4%

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

      5. un-div-inv 48.4%

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

      6. clear-num 48.4%

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

      7. add-sqr-sqrt 48.4%

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

      8. frac-times 48.4%

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

      9. sqrt-unprod 50.2%

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

      10. add-sqr-sqrt 50.3%

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

      11. un-div-inv 50.3%

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

   8. Applied egg-rr 50.3%

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

Alternative 10: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

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 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 0.0)
      (/ c0_m (sqrt (* l (/ V A))))
      (if (<= t_0 5e+306)
        (* c0_m (sqrt 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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m / sqrt((l * (V / A)));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * sqrt(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 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0_m / sqrt((l * (v / a)))
    else if (t_0 <= 5d+306) then
        tmp = c0_m * sqrt(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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m / Math.sqrt((l * (V / A)));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * Math.sqrt(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 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0_m / math.sqrt((l * (V / A)))
	elif t_0 <= 5e+306:
		tmp = c0_m * math.sqrt(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(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0_m / sqrt(Float64(l * Float64(V / A))));
	elseif (t_0 <= 5e+306)
		tmp = Float64(c0_m * sqrt(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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0_m / sqrt((l * (V / A)));
	elseif (t_0 <= 5e+306)
		tmp = c0_m * sqrt(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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[(c0$95$m / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 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 A (*.f64 V l)) < 0.0

   1. Initial program 34.5%

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

      1. associate-/r* 49.8%

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

      2. div-inv 49.8%

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

      3. div-inv 49.8%

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

      4. associate-*l* 38.2%

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

   4. Applied egg-rr 38.2%

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

      1. associate-*l/ 38.2%

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

      2. *-un-lft-identity 38.2%

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

   6. Applied egg-rr 38.2%

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

      1. associate-*r/ 49.8%

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

      2. div-inv 49.8%

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

      3. clear-num 49.8%

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

      4. metadata-eval 49.8%

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

      5. un-div-inv 49.8%

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

      6. clear-num 49.8%

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

      7. add-sqr-sqrt 49.8%

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

      8. frac-times 49.8%

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

      9. sqrt-unprod 49.7%

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

      10. add-sqr-sqrt 49.8%

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

      11. un-div-inv 49.9%

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

   8. Applied egg-rr 49.9%

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

      1. associate-*r/ 34.5%

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

      2. *-commutative 34.5%

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

      3. associate-/l* 49.8%

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

   10. Simplified 49.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 98.7%

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

   if 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 36.9%

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

      1. associate-/r* 48.4%

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

      2. div-inv 48.4%

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

      3. div-inv 48.4%

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

      4. associate-*l* 36.9%

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

   4. Applied egg-rr 36.9%

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

      1. associate-*l/ 36.9%

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

      2. *-un-lft-identity 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. associate-*r/ 48.3%

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

      2. div-inv 48.4%

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

      3. clear-num 48.4%

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

      4. metadata-eval 48.4%

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

      5. un-div-inv 48.4%

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

      6. clear-num 48.4%

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

      7. add-sqr-sqrt 48.4%

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

      8. frac-times 48.4%

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

      9. sqrt-unprod 50.2%

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

      10. add-sqr-sqrt 50.3%

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

      11. un-div-inv 50.3%

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

   8. Applied egg-rr 50.3%

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

Alternative 11: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

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 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 0.0)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 5e+306)
        (* c0_m (sqrt 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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * sqrt(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 = a / (v * l)
    if (t_0 <= 0.0d0) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 5d+306) then
        tmp = c0_m * sqrt(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 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+306) {
		tmp = c0_m * Math.sqrt(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 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+306:
		tmp = c0_m * math.sqrt(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(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+306)
		tmp = Float64(c0_m * sqrt(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 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+306)
		tmp = c0_m * sqrt(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[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+306], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], 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 A (*.f64 V l)) < 0.0

   1. Initial program 34.5%

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

   3. Taylor expanded in c0 around 0 34.5%

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

      1. *-commutative 34.5%

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

      2. associate-/r* 49.8%

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

   5. Simplified 49.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 4.99999999999999993e306

   1. Initial program 98.7%

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

   if 4.99999999999999993e306 < (/.f64 A (*.f64 V l))

   1. Initial program 36.9%

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

      1. associate-/r* 48.4%

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

      2. div-inv 48.4%

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

      3. div-inv 48.4%

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

      4. associate-*l* 36.9%

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

   4. Applied egg-rr 36.9%

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

      1. associate-*l/ 36.9%

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

      2. *-un-lft-identity 36.9%

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

   6. Applied egg-rr 36.9%

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

      1. associate-*r/ 48.3%

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

      2. div-inv 48.4%

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

      3. clear-num 48.4%

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

      4. metadata-eval 48.4%

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

      5. un-div-inv 48.4%

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

      6. clear-num 48.4%

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

      7. add-sqr-sqrt 48.4%

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

      8. frac-times 48.4%

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

      9. sqrt-unprod 50.2%

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

      10. add-sqr-sqrt 50.3%

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

      11. un-div-inv 50.3%

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

   8. Applied egg-rr 50.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

Reproduce

herbie shell --seed 2024135 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))