Henrywood and Agarwal, Equation (3)

Percentage Accurate: 74.0% → 92.0%

Time: 12.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% 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: 92.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-320}:\\ \;\;\;\;c0\_m \cdot \left(\frac{\sqrt{-A}}{\sqrt{-V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+275}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

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) -1e-320)
    (* c0_m (* (/ (sqrt (- A)) (sqrt (- V))) (pow l -0.5)))
    (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+275)))
      (sqrt (* (/ A l) (* c0_m (/ c0_m V))))
      (/ 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) <= -1e-320) {
		tmp = c0_m * ((sqrt(-A) / sqrt(-V)) * pow(l, -0.5));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+275)) {
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	} else {
		tmp = c0_m / (sqrt((V * l)) / sqrt(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) :: tmp
    if ((v * l) <= (-1d-320)) then
        tmp = c0_m * ((sqrt(-a) / sqrt(-v)) * (l ** (-0.5d0)))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 1d+275))) then
        tmp = sqrt(((a / l) * (c0_m * (c0_m / v))))
    else
        tmp = c0_m / (sqrt((v * l)) / sqrt(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 tmp;
	if ((V * l) <= -1e-320) {
		tmp = c0_m * ((Math.sqrt(-A) / Math.sqrt(-V)) * Math.pow(l, -0.5));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+275)) {
		tmp = Math.sqrt(((A / l) * (c0_m * (c0_m / V))));
	} 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) <= -1e-320:
		tmp = c0_m * ((math.sqrt(-A) / math.sqrt(-V)) * math.pow(l, -0.5))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+275):
		tmp = math.sqrt(((A / l) * (c0_m * (c0_m / V))))
	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) <= -1e-320)
		tmp = Float64(c0_m * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) * (l ^ -0.5)));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+275))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m * Float64(c0_m / V))));
	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) <= -1e-320)
		tmp = c0_m * ((sqrt(-A) / sqrt(-V)) * (l ^ -0.5));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+275)))
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	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], -1e-320], N[(c0$95$m * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+275]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m * N[(c0$95$m / V), $MachinePrecision]), $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 3 regimes

2. if (*.f64 V l) < -9.99989e-321

   1. Initial program 77.4%

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

      1. associate-/r* 70.9%

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

      2. clear-num 69.5%

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

      3. sqrt-div 69.5%

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

      4. metadata-eval 69.5%

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

      5. clear-num 69.5%

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

      6. associate-/r* 76.9%

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

      7. clear-num 76.9%

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

      8. associate-/l* 73.8%

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

   4. Applied egg-rr 73.8%

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

      1. /-rgt-identity 73.8%

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

      2. clear-num 73.8%

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

      3. metadata-eval 73.8%

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

      4. sqrt-div 73.7%

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

      5. associate-*r/ 76.8%

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

      6. clear-num 77.4%

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

   6. Applied egg-rr 77.4%

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

      1. remove-double-div 77.4%

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

      2. associate-/r* 70.9%

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

      3. frac-2neg 70.9%

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

      4. distribute-frac-neg2 70.9%

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

      5. sqrt-undiv 43.7%

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

      6. div-inv 43.7%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 5.4%

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

      9. sqr-neg 5.4%

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

      10. sqrt-unprod 3.2%

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

      11. add-sqr-sqrt 3.2%

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

      12. pow1/2 3.2%

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

      13. pow-flip 3.2%

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

      14. add-sqr-sqrt 3.2%

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

      15. sqrt-unprod 27.4%

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

      16. sqr-neg 27.4%

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

      17. sqrt-unprod 38.2%

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

      18. add-sqr-sqrt 38.2%

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

      19. metadata-eval 38.2%

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

   8. Applied egg-rr 38.2%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{V}} \cdot {\ell}^{-0.5}\right)} \]
   9. Step-by-step derivation

      1. frac-2neg 38.2%

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

      2. sqrt-div 49.1%

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

   10. Applied egg-rr 49.1%

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

   if -9.99989e-321 < (*.f64 V l) < -0.0 or 9.9999999999999996e274 < (*.f64 V l)

   1. Initial program 37.1%

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

      1. add-sqr-sqrt 26.4%

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

      2. sqrt-unprod 26.7%

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

      3. *-commutative 26.7%

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

      4. *-commutative 26.7%

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

      5. swap-sqr 25.9%

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

      6. add-sqr-sqrt 25.9%

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

      7. pow2 25.9%

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

   4. Applied egg-rr 25.9%

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

      1. associate-*l/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 31.0%

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

   6. Simplified 31.0%

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

      1. div-inv 31.0%

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

      2. unpow2 31.0%

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

      3. associate-*l* 36.7%

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

   8. Applied egg-rr 36.7%

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

   9. Taylor expanded in c0 around 0 36.7%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999996e274

   1. Initial program 88.5%

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

      1. associate-/r* 78.1%

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

      2. div-inv 78.0%

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

      3. div-inv 78.0%

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

      4. associate-*l* 88.4%

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

   4. Applied egg-rr 88.4%

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

      1. frac-times 88.5%

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

      2. metadata-eval 88.5%

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

      3. div-inv 88.5%

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

      4. clear-num 87.0%

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

      5. associate-*r/ 77.8%

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

      6. sqrt-div 77.7%

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

      7. metadata-eval 77.7%

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

      8. un-div-inv 77.6%

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

      9. associate-*r/ 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. associate-/l* 77.6%

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

   8. Simplified 77.6%

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

      1. associate-*r/ 87.8%

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

      2. sqrt-div 99.4%

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

   10. Applied egg-rr 99.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{-320}:\\ \;\;\;\;c0 \cdot \left(\frac{\sqrt{-A}}{\sqrt{-V}} \cdot {\ell}^{-0.5}\right)\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+275}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 2 \cdot 10^{+276}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 2e+276)))
      (* c0_m (sqrt (/ (/ A V) l)))
      t_0))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+276)) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 2d+276))) then
        tmp = c0_m * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 2e+276)) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 2e+276):
		tmp = c0_m * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 2e+276))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 2e+276)))
		tmp = c0_m * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 2e+276]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 2.0000000000000001e276 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-/l/ 66.4%

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

   3. Simplified 66.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.0000000000000001e276

   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 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+276}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+263}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (<= t_0 0.0)
      (* c0_m (sqrt (/ (/ A V) l)))
      (if (<= t_0 1e+263) t_0 (* c0_m (sqrt (/ (/ A l) V))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+263) {
		tmp = t_0;
	} else {
		tmp = c0_m * sqrt(((A / l) / V));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0_m * sqrt(((a / v) / l))
    else if (t_0 <= 1d+263) then
        tmp = t_0
    else
        tmp = c0_m * sqrt(((a / l) / v))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+263) {
		tmp = t_0;
	} else {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0_m * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+263:
		tmp = t_0
	else:
		tmp = c0_m * math.sqrt(((A / l) / V))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+263)
		tmp = t_0;
	else
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0_m * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+263)
		tmp = t_0;
	else
		tmp = c0_m * sqrt(((A / l) / V));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+263], t$95$0, N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-/l/ 66.3%

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

   3. Simplified 66.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000002e263

   1. Initial program 99.5%

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

   if 1.00000000000000002e263 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 57.8%

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

   3. Taylor expanded in c0 around 0 57.8%

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

      1. *-commutative 57.8%

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

      2. associate-/r* 68.9%

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

   5. Simplified 68.9%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+263}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 71.9%

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

      1. associate-/r* 70.4%

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

      2. div-inv 70.3%

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

      3. div-inv 70.3%

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

      4. associate-*l* 72.2%

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

   4. Applied egg-rr 72.2%

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

      1. frac-times 71.9%

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

      2. metadata-eval 71.9%

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

      3. div-inv 71.9%

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

      4. associate-/l/ 72.7%

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

      5. frac-2neg 72.7%

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

      6. sqrt-div 89.6%

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

      7. distribute-neg-frac2 89.6%

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

   6. Applied egg-rr 89.6%

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

      1. distribute-frac-neg2 89.6%

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

      2. distribute-frac-neg 89.6%

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

   8. Simplified 89.6%

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

   if -4.999999999999985e-310 < V

   1. Initial program 74.5%

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

      1. associate-/r* 72.5%

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

      2. div-inv 72.4%

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

      3. div-inv 72.4%

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

      4. associate-*l* 75.3%

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

   4. Applied egg-rr 75.3%

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

      1. sqrt-prod 43.9%

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

      2. associate-*r* 42.4%

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

      3. *-commutative 42.4%

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

      4. frac-times 41.1%

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

      5. metadata-eval 41.1%

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

      6. sqrt-div 41.1%

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

      7. metadata-eval 41.1%

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

      8. div-inv 41.1%

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

      9. sqrt-prod 47.8%

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

      10. associate-/r* 47.5%

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

      11. *-commutative 47.5%

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

   6. Applied egg-rr 47.5%

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

4. Final simplification 68.7%

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

Alternative 5: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+291}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-320}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+275}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

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) -5e+291)
    (* c0_m (/ (sqrt (/ A (- l))) (sqrt (- V))))
    (if (<= (* V l) -1e-320)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+275)))
        (sqrt (* (/ A l) (* c0_m (/ c0_m V))))
        (/ 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) <= -5e+291) {
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	} else if ((V * l) <= -1e-320) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+275)) {
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	} else {
		tmp = c0_m / (sqrt((V * l)) / sqrt(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) :: tmp
    if ((v * l) <= (-5d+291)) then
        tmp = c0_m * (sqrt((a / -l)) / sqrt(-v))
    else if ((v * l) <= (-1d-320)) then
        tmp = c0_m * (sqrt(-a) / sqrt((v * -l)))
    else if (((v * l) <= 0.0d0) .or. (.not. ((v * l) <= 1d+275))) then
        tmp = sqrt(((a / l) * (c0_m * (c0_m / v))))
    else
        tmp = c0_m / (sqrt((v * l)) / sqrt(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 tmp;
	if ((V * l) <= -5e+291) {
		tmp = c0_m * (Math.sqrt((A / -l)) / Math.sqrt(-V));
	} else if ((V * l) <= -1e-320) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+275)) {
		tmp = Math.sqrt(((A / l) * (c0_m * (c0_m / V))));
	} 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) <= -5e+291:
		tmp = c0_m * (math.sqrt((A / -l)) / math.sqrt(-V))
	elif (V * l) <= -1e-320:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+275):
		tmp = math.sqrt(((A / l) * (c0_m * (c0_m / V))))
	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) <= -5e+291)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / Float64(-l))) / sqrt(Float64(-V))));
	elseif (Float64(V * l) <= -1e-320)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+275))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m * Float64(c0_m / V))));
	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) <= -5e+291)
		tmp = c0_m * (sqrt((A / -l)) / sqrt(-V));
	elseif ((V * l) <= -1e-320)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+275)))
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	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], -5e+291], N[(c0$95$m * N[(N[Sqrt[N[(A / (-l)), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-320], 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+275]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m * N[(c0$95$m / V), $MachinePrecision]), $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) < -5.0000000000000001e291

   1. Initial program 38.4%

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

      1. associate-/r* 65.3%

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

      2. div-inv 65.2%

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

      3. div-inv 65.2%

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

      4. associate-*l* 38.6%

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

   4. Applied egg-rr 38.6%

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

      1. frac-times 38.4%

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

      2. metadata-eval 38.4%

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

      3. div-inv 38.4%

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

      4. associate-/l/ 65.1%

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

      5. frac-2neg 65.1%

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

      6. sqrt-div 49.8%

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

      7. distribute-neg-frac2 49.8%

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

   6. Applied egg-rr 49.8%

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

      1. distribute-frac-neg2 49.8%

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

      2. distribute-frac-neg 49.8%

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

   8. Simplified 49.8%

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

   if -5.0000000000000001e291 < (*.f64 V l) < -9.99989e-321

   1. Initial program 83.0%

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

      1. frac-2neg 83.0%

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

      2. sqrt-div 98.8%

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

      3. *-commutative 98.8%

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

      4. distribute-rgt-neg-in 98.8%

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

   4. Applied egg-rr 98.8%

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

   if -9.99989e-321 < (*.f64 V l) < -0.0 or 9.9999999999999996e274 < (*.f64 V l)

   1. Initial program 37.1%

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

      1. add-sqr-sqrt 26.4%

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

      2. sqrt-unprod 26.7%

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

      3. *-commutative 26.7%

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

      4. *-commutative 26.7%

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

      5. swap-sqr 25.9%

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

      6. add-sqr-sqrt 25.9%

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

      7. pow2 25.9%

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

   4. Applied egg-rr 25.9%

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

      1. associate-*l/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 31.0%

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

   6. Simplified 31.0%

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

      1. div-inv 31.0%

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

      2. unpow2 31.0%

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

      3. associate-*l* 36.7%

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

   8. Applied egg-rr 36.7%

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

   9. Taylor expanded in c0 around 0 36.7%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999996e274

   1. Initial program 88.5%

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

      1. associate-/r* 78.1%

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

      2. div-inv 78.0%

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

      3. div-inv 78.0%

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

      4. associate-*l* 88.4%

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

   4. Applied egg-rr 88.4%

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

      1. frac-times 88.5%

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

      2. metadata-eval 88.5%

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

      3. div-inv 88.5%

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

      4. clear-num 87.0%

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

      5. associate-*r/ 77.8%

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

      6. sqrt-div 77.7%

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

      7. metadata-eval 77.7%

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

      8. un-div-inv 77.6%

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

      9. associate-*r/ 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. associate-/l* 77.6%

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

   8. Simplified 77.6%

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

      1. associate-*r/ 87.8%

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

      2. sqrt-div 99.4%

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

   10. Applied egg-rr 99.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+291}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{-\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-320}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+275}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0\_m}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-320}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+275}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \end{array} \]

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))
    (* (sqrt (/ A V)) (/ c0_m (sqrt l)))
    (if (<= (* V l) -1e-320)
      (* c0_m (/ (sqrt (- A)) (sqrt (* V (- l)))))
      (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+275)))
        (sqrt (* (/ A l) (* c0_m (/ c0_m V))))
        (/ 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 = sqrt((A / V)) * (c0_m / sqrt(l));
	} else if ((V * l) <= -1e-320) {
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+275)) {
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	} 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 = Math.sqrt((A / V)) * (c0_m / Math.sqrt(l));
	} else if ((V * l) <= -1e-320) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+275)) {
		tmp = Math.sqrt(((A / l) * (c0_m * (c0_m / V))));
	} 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 = math.sqrt((A / V)) * (c0_m / math.sqrt(l))
	elif (V * l) <= -1e-320:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+275):
		tmp = math.sqrt(((A / l) * (c0_m * (c0_m / V))))
	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(sqrt(Float64(A / V)) * Float64(c0_m / sqrt(l)));
	elseif (Float64(V * l) <= -1e-320)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+275))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m * Float64(c0_m / V))));
	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 = sqrt((A / V)) * (c0_m / sqrt(l));
	elseif ((V * l) <= -1e-320)
		tmp = c0_m * (sqrt(-A) / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+275)))
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	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[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-320], 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+275]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m * N[(c0$95$m / V), $MachinePrecision]), $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 21.9%

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

      1. associate-/r* 56.0%

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

      2. sqrt-div 36.3%

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

      3. associate-*r/ 36.4%

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

   4. Applied egg-rr 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-/l* 36.7%

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

   6. Simplified 36.7%

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

   if -inf.0 < (*.f64 V l) < -9.99989e-321

   1. Initial program 83.4%

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

      1. frac-2neg 83.4%

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

      2. sqrt-div 98.8%

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

      3. *-commutative 98.8%

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

      4. distribute-rgt-neg-in 98.8%

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

   4. Applied egg-rr 98.8%

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

   if -9.99989e-321 < (*.f64 V l) < -0.0 or 9.9999999999999996e274 < (*.f64 V l)

   1. Initial program 37.1%

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

      1. add-sqr-sqrt 26.4%

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

      2. sqrt-unprod 26.7%

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

      3. *-commutative 26.7%

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

      4. *-commutative 26.7%

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

      5. swap-sqr 25.9%

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

      6. add-sqr-sqrt 25.9%

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

      7. pow2 25.9%

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

   4. Applied egg-rr 25.9%

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

      1. associate-*l/ 25.8%

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

      2. *-commutative 25.8%

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

      3. times-frac 31.0%

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

   6. Simplified 31.0%

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

      1. div-inv 31.0%

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

      2. unpow2 31.0%

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

      3. associate-*l* 36.7%

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

   8. Applied egg-rr 36.7%

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

   9. Taylor expanded in c0 around 0 36.7%

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

   if -0.0 < (*.f64 V l) < 9.9999999999999996e274

   1. Initial program 88.5%

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

      1. associate-/r* 78.1%

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

      2. div-inv 78.0%

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

      3. div-inv 78.0%

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

      4. associate-*l* 88.4%

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

   4. Applied egg-rr 88.4%

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

      1. frac-times 88.5%

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

      2. metadata-eval 88.5%

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

      3. div-inv 88.5%

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

      4. clear-num 87.0%

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

      5. associate-*r/ 77.8%

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

      6. sqrt-div 77.7%

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

      7. metadata-eval 77.7%

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

      8. un-div-inv 77.6%

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

      9. associate-*r/ 87.8%

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

   6. Applied egg-rr 87.8%

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

      1. associate-/l* 77.6%

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

   8. Simplified 77.6%

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

      1. associate-*r/ 87.8%

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

      2. sqrt-div 99.4%

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

   10. Applied egg-rr 99.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-320}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+275}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{V \cdot \ell}}{\sqrt{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-315}:\\ \;\;\;\;c0\_m \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+298}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 1e-315)
      (* c0_m (* (pow l -0.5) (sqrt (/ A V))))
      (if (<= t_0 1e+298)
        (* c0_m (sqrt t_0))
        (sqrt (* (/ A l) (* c0_m (/ c0_m V)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = c0_m * (pow(l, -0.5) * sqrt((A / V)));
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 1d-315) then
        tmp = c0_m * ((l ** (-0.5d0)) * sqrt((a / v)))
    else if (t_0 <= 1d+298) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = sqrt(((a / l) * (c0_m * (c0_m / v))))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = c0_m * (Math.pow(l, -0.5) * Math.sqrt((A / V)));
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(((A / l) * (c0_m * (c0_m / V))));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 1e-315:
		tmp = c0_m * (math.pow(l, -0.5) * math.sqrt((A / V)))
	elif t_0 <= 1e+298:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = math.sqrt(((A / l) * (c0_m * (c0_m / V))))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 1e-315)
		tmp = Float64(c0_m * Float64((l ^ -0.5) * sqrt(Float64(A / V))));
	elseif (t_0 <= 1e+298)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m * Float64(c0_m / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-315)
		tmp = c0_m * ((l ^ -0.5) * sqrt((A / V)));
	elseif (t_0 <= 1e+298)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 1e-315], N[(c0$95$m * N[(N[Power[l, -0.5], $MachinePrecision] * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+298], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 9.999999985e-316

   1. Initial program 34.1%

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

      1. associate-/r* 49.6%

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

      2. clear-num 46.7%

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

      3. sqrt-div 46.6%

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

      4. metadata-eval 46.6%

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

      5. clear-num 46.6%

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

      6. associate-/r* 32.0%

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

      7. clear-num 32.0%

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

      8. associate-/l* 46.6%

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

   4. Applied egg-rr 46.6%

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

      1. /-rgt-identity 46.6%

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

      2. clear-num 46.6%

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

      3. metadata-eval 46.6%

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

      4. sqrt-div 46.6%

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

      5. associate-*r/ 32.0%

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

      6. clear-num 34.1%

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

   6. Applied egg-rr 34.1%

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

      1. remove-double-div 34.1%

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

      2. associate-/r* 49.6%

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

      3. frac-2neg 49.6%

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

      4. distribute-frac-neg2 49.6%

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

      5. sqrt-undiv 34.8%

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

      6. div-inv 34.9%

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

      7. add-sqr-sqrt 13.5%

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

      8. sqrt-unprod 16.3%

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

      9. sqr-neg 16.3%

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

      10. sqrt-unprod 6.1%

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

      11. add-sqr-sqrt 12.0%

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

      12. pow1/2 12.0%

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

      13. pow-flip 12.0%

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

      14. add-sqr-sqrt 12.0%

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

      15. sqrt-unprod 34.3%

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

      16. sqr-neg 34.3%

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

      17. sqrt-unprod 40.8%

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

      18. add-sqr-sqrt 40.8%

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

      19. metadata-eval 40.8%

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

   8. Applied egg-rr 40.8%

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

   if 9.999999985e-316 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e297

   1. Initial program 99.5%

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

   if 9.9999999999999996e297 < (/.f64 A (*.f64 V l))

   1. Initial program 35.4%

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

      1. add-sqr-sqrt 22.5%

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

      2. sqrt-unprod 22.7%

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

      3. *-commutative 22.7%

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

      4. *-commutative 22.7%

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

      5. swap-sqr 22.1%

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

      6. add-sqr-sqrt 22.1%

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

      7. pow2 22.1%

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

   4. Applied egg-rr 22.1%

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

      1. associate-*l/ 24.8%

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

      2. *-commutative 24.8%

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

      3. times-frac 28.5%

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

   6. Simplified 28.5%

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

      1. div-inv 28.5%

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

      2. unpow2 28.5%

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

      3. associate-*l* 34.0%

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

   8. Applied egg-rr 34.0%

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

   9. Taylor expanded in c0 around 0 34.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 10^{-315}:\\ \;\;\;\;c0 \cdot \left({\ell}^{-0.5} \cdot \sqrt{\frac{A}{V}}\right)\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+298}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-315}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0\_m}{\sqrt{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+298}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 1e-315)
      (* (sqrt (/ A V)) (/ c0_m (sqrt l)))
      (if (<= t_0 1e+298)
        (* c0_m (sqrt t_0))
        (sqrt (* (/ A l) (* c0_m (/ c0_m V)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = sqrt((A / V)) * (c0_m / sqrt(l));
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 1d-315) then
        tmp = sqrt((a / v)) * (c0_m / sqrt(l))
    else if (t_0 <= 1d+298) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = sqrt(((a / l) * (c0_m * (c0_m / v))))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = Math.sqrt((A / V)) * (c0_m / Math.sqrt(l));
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(((A / l) * (c0_m * (c0_m / V))));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 1e-315:
		tmp = math.sqrt((A / V)) * (c0_m / math.sqrt(l))
	elif t_0 <= 1e+298:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = math.sqrt(((A / l) * (c0_m * (c0_m / V))))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 1e-315)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0_m / sqrt(l)));
	elseif (t_0 <= 1e+298)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m * Float64(c0_m / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-315)
		tmp = sqrt((A / V)) * (c0_m / sqrt(l));
	elseif (t_0 <= 1e+298)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 1e-315], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0$95$m / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+298], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 9.999999985e-316

   1. Initial program 34.1%

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

      1. associate-/r* 49.6%

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

      2. sqrt-div 40.8%

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

      3. associate-*r/ 40.8%

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

   4. Applied egg-rr 40.8%

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

      1. *-commutative 40.8%

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

      2. associate-/l* 40.9%

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

   6. Simplified 40.9%

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

   if 9.999999985e-316 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e297

   1. Initial program 99.5%

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

   if 9.9999999999999996e297 < (/.f64 A (*.f64 V l))

   1. Initial program 35.4%

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

      1. add-sqr-sqrt 22.5%

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

      2. sqrt-unprod 22.7%

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

      3. *-commutative 22.7%

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

      4. *-commutative 22.7%

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

      5. swap-sqr 22.1%

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

      6. add-sqr-sqrt 22.1%

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

      7. pow2 22.1%

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

   4. Applied egg-rr 22.1%

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

      1. associate-*l/ 24.8%

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

      2. *-commutative 24.8%

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

      3. times-frac 28.5%

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

   6. Simplified 28.5%

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

      1. div-inv 28.5%

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

      2. unpow2 28.5%

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

      3. associate-*l* 34.0%

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

   8. Applied egg-rr 34.0%

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

   9. Taylor expanded in c0 around 0 34.0%

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

Alternative 9: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{-315}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+298}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 1e-315)
      (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
      (if (<= t_0 1e+298)
        (* c0_m (sqrt t_0))
        (sqrt (* (/ A l) (* c0_m (/ c0_m V)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 1d-315) then
        tmp = c0_m * (sqrt((a / v)) / sqrt(l))
    else if (t_0 <= 1d+298) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = sqrt(((a / l) * (c0_m * (c0_m / v))))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 1e-315) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(((A / l) * (c0_m * (c0_m / V))));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 1e-315:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif t_0 <= 1e+298:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = math.sqrt(((A / l) * (c0_m * (c0_m / V))))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 1e-315)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (t_0 <= 1e+298)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m * Float64(c0_m / V))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 1e-315)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif (t_0 <= 1e+298)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 1e-315], N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+298], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 9.999999985e-316

   1. Initial program 34.1%

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

      1. associate-/r* 49.6%

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

      2. sqrt-div 40.8%

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

      3. associate-*r/ 40.8%

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

   4. Applied egg-rr 40.8%

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

      1. associate-/l* 40.8%

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

   6. Simplified 40.8%

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

   if 9.999999985e-316 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e297

   1. Initial program 99.5%

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

   if 9.9999999999999996e297 < (/.f64 A (*.f64 V l))

   1. Initial program 35.4%

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

      1. add-sqr-sqrt 22.5%

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

      2. sqrt-unprod 22.7%

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

      3. *-commutative 22.7%

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

      4. *-commutative 22.7%

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

      5. swap-sqr 22.1%

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

      6. add-sqr-sqrt 22.1%

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

      7. pow2 22.1%

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

   4. Applied egg-rr 22.1%

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

      1. associate-*l/ 24.8%

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

      2. *-commutative 24.8%

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

      3. times-frac 28.5%

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

   6. Simplified 28.5%

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

      1. div-inv 28.5%

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

      2. unpow2 28.5%

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

      3. associate-*l* 34.0%

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

   8. Applied egg-rr 34.0%

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

   9. Taylor expanded in c0 around 0 34.0%

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

Alternative 10: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 10^{+298}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

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 1e+298)))
      (sqrt (* (/ A l) (* c0_m (/ c0_m 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 <= 1e+298)) {
		tmp = sqrt(((A / l) * (c0_m * (c0_m / 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 <= 1d+298))) then
        tmp = sqrt(((a / l) * (c0_m * (c0_m / 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 <= 1e+298)) {
		tmp = Math.sqrt(((A / l) * (c0_m * (c0_m / 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 <= 1e+298):
		tmp = math.sqrt(((A / l) * (c0_m * (c0_m / 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 <= 1e+298))
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m * Float64(c0_m / 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 <= 1e+298)))
		tmp = sqrt(((A / l) * (c0_m * (c0_m / 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, 1e+298]], $MachinePrecision]], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m * N[(c0$95$m / V), $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 9.9999999999999996e297 < (/.f64 A (*.f64 V l))

   1. Initial program 33.6%

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

      1. add-sqr-sqrt 26.9%

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

      2. sqrt-unprod 27.0%

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

      3. *-commutative 27.0%

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

      4. *-commutative 27.0%

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

      5. swap-sqr 26.2%

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

      6. add-sqr-sqrt 26.3%

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

      7. pow2 26.3%

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

   4. Applied egg-rr 26.3%

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

      1. associate-*l/ 29.6%

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

      2. *-commutative 29.6%

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

      3. times-frac 31.9%

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

   6. Simplified 31.9%

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

      1. div-inv 31.9%

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

      2. unpow2 31.9%

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

      3. associate-*l* 36.8%

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

   8. Applied egg-rr 36.8%

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

   9. Taylor expanded in c0 around 0 36.8%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e297

   1. Initial program 99.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 10^{+298}\right):\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{c0\_m}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}\\ \mathbf{elif}\;t\_0 \leq 10^{+298}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \end{array} \]

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 (pow (/ (/ A V) l) -0.5))
      (if (<= t_0 1e+298)
        (* c0_m (sqrt t_0))
        (/ c0_m (sqrt (* l (/ V 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 / pow(((A / V) / l), -0.5);
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = c0_m / sqrt((l * (V / 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 / (((a / v) / l) ** (-0.5d0))
    else if (t_0 <= 1d+298) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = c0_m / sqrt((l * (v / 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.pow(((A / V) / l), -0.5);
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = c0_m / Math.sqrt((l * (V / 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.pow(((A / V) / l), -0.5)
	elif t_0 <= 1e+298:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = c0_m / math.sqrt((l * (V / 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 / (Float64(Float64(A / V) / l) ^ -0.5));
	elseif (t_0 <= 1e+298)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = Float64(c0_m / sqrt(Float64(l * Float64(V / 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 / (((A / V) / l) ^ -0.5);
	elseif (t_0 <= 1e+298)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = c0_m / sqrt((l * (V / 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[Power[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+298], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l * N[(V / 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 31.7%

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

      1. associate-/r* 49.1%

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

      2. clear-num 47.3%

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

      3. sqrt-div 47.2%

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

      4. metadata-eval 47.2%

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

      5. clear-num 47.2%

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

      6. associate-/r* 31.7%

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

      7. clear-num 31.7%

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

      8. associate-/l* 47.2%

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

   4. Applied egg-rr 47.2%

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

      1. /-rgt-identity 47.2%

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

      2. clear-num 47.2%

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

      3. metadata-eval 47.2%

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

      4. sqrt-div 47.2%

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

      5. associate-*r/ 31.7%

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

      6. clear-num 31.7%

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

   6. Applied egg-rr 31.7%

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

      1. pow1/2 31.7%

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

      2. pow-flip 31.7%

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

      3. metadata-eval 31.7%

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

   8. Applied egg-rr 31.7%

      \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}} \]
   9. Step-by-step derivation

      1. associate-/r* 49.1%

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

   10. Simplified 49.1%

       \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}} \]
   11. Step-by-step derivation

       1. un-div-inv 49.2%

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

       2. sqr-pow 49.1%

          \[\leadsto \frac{c0}{\color{blue}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\frac{\frac{A}{V}}{\ell}\right)}^{\left(\frac{-0.5}{2}\right)}}} \]

       3. associate-/r* 49.1%

          \[\leadsto \color{blue}{\frac{\frac{c0}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{\left(\frac{-0.5}{2}\right)}}}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{\left(\frac{-0.5}{2}\right)}}} \]

       4. associate-/r* 31.7%

          \[\leadsto \frac{\frac{c0}{{\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{\left(\frac{-0.5}{2}\right)}}}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{\left(\frac{-0.5}{2}\right)}} \]

       5. metadata-eval 31.7%

          \[\leadsto \frac{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{-0.25}}}}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{\left(\frac{-0.5}{2}\right)}} \]

       6. associate-/r* 31.7%

          \[\leadsto \frac{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.25}}}{{\color{blue}{\left(\frac{A}{V \cdot \ell}\right)}}^{\left(\frac{-0.5}{2}\right)}} \]

       7. metadata-eval 31.7%

          \[\leadsto \frac{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.25}}}{{\left(\frac{A}{V \cdot \ell}\right)}^{\color{blue}{-0.25}}} \]

   12. Applied egg-rr 31.7%

       \[\leadsto \color{blue}{\frac{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.25}}}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.25}}} \]
   13. Step-by-step derivation

       1. associate-/l/ 31.7%

          \[\leadsto \color{blue}{\frac{c0}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.25} \cdot {\left(\frac{A}{V \cdot \ell}\right)}^{-0.25}}} \]

       2. pow-sqr 31.7%

          \[\leadsto \frac{c0}{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{\left(2 \cdot -0.25\right)}}} \]

       3. metadata-eval 31.7%

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

       4. associate-/r* 49.2%

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

   14. Simplified 49.2%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e297

   1. Initial program 99.0%

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

   if 9.9999999999999996e297 < (/.f64 A (*.f64 V l))

   1. Initial program 35.4%

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

      1. associate-/r* 50.1%

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

      2. clear-num 50.1%

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

      3. sqrt-div 51.5%

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

      4. metadata-eval 51.5%

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

      5. clear-num 50.0%

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

      6. associate-/r* 35.4%

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

      7. clear-num 36.8%

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

      8. associate-/l* 50.0%

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

   4. Applied egg-rr 50.0%

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

      1. /-rgt-identity 50.0%

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

      2. clear-num 50.0%

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

      3. metadata-eval 50.0%

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

      4. sqrt-div 50.0%

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

      5. associate-*r/ 35.4%

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

      6. clear-num 35.4%

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

   6. Applied egg-rr 35.4%

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

      1. pow1/2 35.4%

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

      2. pow-flip 35.4%

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

      3. metadata-eval 35.4%

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

   8. Applied egg-rr 35.4%

      \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}} \]
   9. Step-by-step derivation

      1. associate-/r* 50.0%

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

   10. Simplified 50.0%

       \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}} \]
   11. Step-by-step derivation

       1. un-div-inv 50.0%

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

       2. add-sqr-sqrt 49.9%

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

       3. sqrt-unprod 50.0%

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

       4. pow-prod-up 50.0%

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

       5. metadata-eval 50.0%

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

       6. inv-pow 50.0%

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

       7. clear-num 51.5%

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

       8. div-inv 51.5%

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

       9. clear-num 51.5%

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

   12. Applied egg-rr 51.5%

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

Alternative 12: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t\_0 \leq 10^{+298}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \end{array} \]

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 V) l)))
      (if (<= t_0 1e+298)
        (* c0_m (sqrt t_0))
        (/ c0_m (sqrt (* l (/ V 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 / V) / l));
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * sqrt(t_0);
	} else {
		tmp = c0_m / sqrt((l * (V / 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 / v) / l))
    else if (t_0 <= 1d+298) then
        tmp = c0_m * sqrt(t_0)
    else
        tmp = c0_m / sqrt((l * (v / 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 / V) / l));
	} else if (t_0 <= 1e+298) {
		tmp = c0_m * Math.sqrt(t_0);
	} else {
		tmp = c0_m / Math.sqrt((l * (V / 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 / V) / l))
	elif t_0 <= 1e+298:
		tmp = c0_m * math.sqrt(t_0)
	else:
		tmp = c0_m / math.sqrt((l * (V / 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 / V) / l)));
	elseif (t_0 <= 1e+298)
		tmp = Float64(c0_m * sqrt(t_0));
	else
		tmp = Float64(c0_m / sqrt(Float64(l * Float64(V / 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 / V) / l));
	elseif (t_0 <= 1e+298)
		tmp = c0_m * sqrt(t_0);
	else
		tmp = c0_m / sqrt((l * (V / 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 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+298], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(l * N[(V / 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 31.7%

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

      1. *-commutative 31.7%

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

      2. associate-/l/ 49.1%

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

   3. Simplified 49.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 9.9999999999999996e297

   1. Initial program 99.0%

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

   if 9.9999999999999996e297 < (/.f64 A (*.f64 V l))

   1. Initial program 35.4%

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

      1. associate-/r* 50.1%

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

      2. clear-num 50.1%

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

      3. sqrt-div 51.5%

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

      4. metadata-eval 51.5%

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

      5. clear-num 50.0%

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

      6. associate-/r* 35.4%

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

      7. clear-num 36.8%

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

      8. associate-/l* 50.0%

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

   4. Applied egg-rr 50.0%

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

      1. /-rgt-identity 50.0%

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

      2. clear-num 50.0%

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

      3. metadata-eval 50.0%

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

      4. sqrt-div 50.0%

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

      5. associate-*r/ 35.4%

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

      6. clear-num 35.4%

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

   6. Applied egg-rr 35.4%

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

      1. pow1/2 35.4%

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

      2. pow-flip 35.4%

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

      3. metadata-eval 35.4%

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

   8. Applied egg-rr 35.4%

      \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(\frac{A}{V \cdot \ell}\right)}^{-0.5}}} \]
   9. Step-by-step derivation

      1. associate-/r* 50.0%

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

   10. Simplified 50.0%

       \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(\frac{\frac{A}{V}}{\ell}\right)}^{-0.5}}} \]
   11. Step-by-step derivation

       1. un-div-inv 50.0%

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

       2. add-sqr-sqrt 49.9%

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

       3. sqrt-unprod 50.0%

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

       4. pow-prod-up 50.0%

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

       5. metadata-eval 50.0%

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

       6. inv-pow 50.0%

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

       7. clear-num 51.5%

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

       8. div-inv 51.5%

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

       9. clear-num 51.5%

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

   12. Applied egg-rr 51.5%

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

Alternative 13: 74.0% 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 73.2%

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

Reproduce

herbie shell --seed 2024167 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))