Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.2% → 90.7%

Time: 10.8s

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: 73.2% 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: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;{\ell}^{-0.5} \cdot \left(c0\_m \cdot \sqrt{\frac{A}{V}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-284}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-299}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+302}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) (- INFINITY))
    (* (pow l -0.5) (* c0_m (sqrt (/ A V))))
    (if (<= (* V l) -5e-284)
      (* c0_m (/ (sqrt (- A)) (sqrt (* l (- V)))))
      (if (<= (* V l) 1e-299)
        (* c0_m (sqrt (/ (/ A l) V)))
        (if (<= (* V l) 2e+302)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (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 tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = pow(l, -0.5) * (c0_m * sqrt((A / V)));
	} else if ((V * l) <= -5e-284) {
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	} else if ((V * l) <= 1e-299) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if ((V * l) <= 2e+302) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	}
	return c0_s * tmp;
}

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.pow(l, -0.5) * (c0_m * Math.sqrt((A / V)));
	} else if ((V * l) <= -5e-284) {
		tmp = c0_m * (Math.sqrt(-A) / Math.sqrt((l * -V)));
	} else if ((V * l) <= 1e-299) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if ((V * l) <= 2e+302) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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):
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = math.pow(l, -0.5) * (c0_m * math.sqrt((A / V)))
	elif (V * l) <= -5e-284:
		tmp = c0_m * (math.sqrt(-A) / math.sqrt((l * -V)))
	elif (V * l) <= 1e-299:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif (V * l) <= 2e+302:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	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)
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64((l ^ -0.5) * Float64(c0_m * sqrt(Float64(A / V))));
	elseif (Float64(V * l) <= -5e-284)
		tmp = Float64(c0_m * Float64(sqrt(Float64(-A)) / sqrt(Float64(l * Float64(-V)))));
	elseif (Float64(V * l) <= 1e-299)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(V * l) <= 2e+302)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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)
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = (l ^ -0.5) * (c0_m * sqrt((A / V)));
	elseif ((V * l) <= -5e-284)
		tmp = c0_m * (sqrt(-A) / sqrt((l * -V)));
	elseif ((V * l) <= 1e-299)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif ((V * l) <= 2e+302)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	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_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(N[Power[l, -0.5], $MachinePrecision] * N[(c0$95$m * N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-284], N[(c0$95$m * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(l * (-V)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-299], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+302], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $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 5 regimes

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

   1. Initial program 31.1%

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

      1. *-commutative 31.1%

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

      2. associate-/r* 45.0%

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

      3. sqrt-div 71.2%

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

      4. associate-*l/ 71.3%

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

   4. Applied egg-rr 71.3%

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

      1. clear-num 70.9%

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

      2. associate-/r/ 71.4%

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

      3. pow1/2 71.4%

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

      4. pow-flip 71.3%

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

      5. metadata-eval 71.3%

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

      6. *-commutative 71.3%

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

   6. Applied egg-rr 71.3%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999973e-284

   1. Initial program 84.7%

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

      1. frac-2neg 84.7%

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

      2. sqrt-div 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   4. Applied egg-rr 99.5%

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

   if -4.99999999999999973e-284 < (*.f64 V l) < 9.99999999999999992e-300

   1. Initial program 49.8%

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

   3. Taylor expanded in c0 around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. associate-/r* 79.6%

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

   5. Simplified 79.6%

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

   if 9.99999999999999992e-300 < (*.f64 V l) < 2.0000000000000002e302

   1. Initial program 81.2%

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

      1. sqrt-div 99.4%

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

      2. associate-*r/ 98.0%

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

   4. Applied egg-rr 98.0%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   if 2.0000000000000002e302 < (*.f64 V l)

   1. Initial program 38.6%

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

      1. add-sqr-sqrt 38.6%

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

      2. sqrt-unprod 38.6%

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

      3. *-commutative 38.6%

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

      4. *-commutative 38.6%

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

      5. swap-sqr 37.4%

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

      6. add-sqr-sqrt 37.4%

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

      7. pow2 37.4%

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

   4. Applied egg-rr 37.4%

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

      1. associate-*l/ 37.1%

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

      2. *-commutative 37.1%

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

      3. times-frac 42.3%

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

   6. Simplified 42.3%

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

      1. unpow2 42.3%

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

      2. associate-/l* 47.1%

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

   8. Applied egg-rr 47.1%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;{\ell}^{-0.5} \cdot \left(c0 \cdot \sqrt{\frac{A}{V}}\right)\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-284}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{\ell \cdot \left(-V\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-299}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 4 \cdot 10^{+278}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 4e+278)))
      (* c0_m (sqrt (/ (/ A l) V)))
      t_0))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+278)) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 4d+278))) then
        tmp = c0_m * sqrt(((a / l) / v))
    else
        tmp = t_0
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 4e+278)) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else {
		tmp = t_0;
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 4e+278):
		tmp = c0_m * math.sqrt(((A / l) / V))
	else:
		tmp = t_0
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 4e+278))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	else
		tmp = t_0;
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 4e+278)))
		tmp = c0_m * sqrt(((A / l) / V));
	else
		tmp = t_0;
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 4e+278]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $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 3.99999999999999985e278 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 61.8%

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

   3. Taylor expanded in c0 around 0 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-/r* 67.8%

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

   5. Simplified 67.8%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999985e278

   1. Initial program 98.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+278}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+278}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \end{array} \]

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 l) V)))
      (if (<= t_0 4e+278) t_0 (/ c0_m (sqrt (/ V (/ A 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 t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 4e+278) {
		tmp = t_0;
	} else {
		tmp = c0_m / sqrt((V / (A / 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) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 4d+278) then
        tmp = t_0
    else
        tmp = c0_m / sqrt((v / (a / 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 t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 4e+278) {
		tmp = t_0;
	} else {
		tmp = c0_m / Math.sqrt((V / (A / 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):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 4e+278:
		tmp = t_0
	else:
		tmp = c0_m / math.sqrt((V / (A / 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)
	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 / l) / V)));
	elseif (t_0 <= 4e+278)
		tmp = t_0;
	else
		tmp = Float64(c0_m / sqrt(Float64(V / Float64(A / 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)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 4e+278)
		tmp = t_0;
	else
		tmp = c0_m / sqrt((V / (A / 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_] := 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 / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+278], t$95$0, N[(c0$95$m / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.7%

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

   3. Taylor expanded in c0 around 0 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-/r* 71.3%

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

   5. Simplified 71.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999985e278

   1. Initial program 98.5%

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

   if 3.99999999999999985e278 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 45.8%

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

      1. *-un-lft-identity 45.8%

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

      2. times-frac 53.7%

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

   4. Applied egg-rr 53.7%

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

      1. frac-times 45.8%

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

      2. *-un-lft-identity 45.8%

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

      3. sqrt-undiv 43.5%

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

      4. clear-num 43.5%

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

      5. un-div-inv 43.5%

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

      6. sqrt-undiv 48.6%

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

   6. Applied egg-rr 48.6%

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

      1. associate-*r/ 55.2%

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

   8. Simplified 55.2%

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

      1. associate-*r/ 48.6%

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

      2. *-un-lft-identity 48.6%

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

      3. times-frac 55.2%

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

      4. associate-/r/ 55.2%

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

      5. clear-num 55.2%

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

   10. Applied egg-rr 55.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+278}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+278}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (* c0_m (sqrt (/ A (* V l))))))
   (*
    c0_s
    (if (<= t_0 0.0)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 4e+278) t_0 (/ c0_m (sqrt (* V (/ l A)))))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 4e+278) {
		tmp = t_0;
	} else {
		tmp = c0_m / sqrt((V * (l / A)));
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0_m * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 4d+278) then
        tmp = t_0
    else
        tmp = c0_m / sqrt((v * (l / a)))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = c0_m * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 4e+278) {
		tmp = t_0;
	} else {
		tmp = c0_m / Math.sqrt((V * (l / A)));
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = c0_m * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 4e+278:
		tmp = t_0
	else:
		tmp = c0_m / math.sqrt((V * (l / A)))
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(c0_m * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 4e+278)
		tmp = t_0;
	else
		tmp = Float64(c0_m / sqrt(Float64(V * Float64(l / A))));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = c0_m * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 4e+278)
		tmp = t_0;
	else
		tmp = c0_m / sqrt((V * (l / A)));
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 0.0], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+278], t$95$0, N[(c0$95$m / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.7%

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

   3. Taylor expanded in c0 around 0 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-/r* 71.3%

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

   5. Simplified 71.3%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 3.99999999999999985e278

   1. Initial program 98.5%

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

   if 3.99999999999999985e278 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 45.8%

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

      1. *-un-lft-identity 45.8%

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

      2. times-frac 53.7%

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

   4. Applied egg-rr 53.7%

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

      1. frac-times 45.8%

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

      2. *-un-lft-identity 45.8%

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

      3. sqrt-undiv 43.5%

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

      4. clear-num 43.5%

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

      5. un-div-inv 43.5%

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

      6. sqrt-undiv 48.6%

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

   6. Applied egg-rr 48.6%

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

      1. associate-*r/ 55.2%

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

   8. Simplified 55.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 4 \cdot 10^{+278}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+182}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-58}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-299}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+302}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) -2e+182)
    (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
    (if (<= (* V l) -2e-58)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= (* V l) -5e-284)
        (/ c0_m (* (sqrt l) (sqrt (/ V A))))
        (if (<= (* V l) 1e-299)
          (* c0_m (sqrt (/ (/ A l) V)))
          (if (<= (* V l) 2e+302)
            (* c0_m (/ (sqrt A) (sqrt (* V l))))
            (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 tmp;
	if ((V * l) <= -2e+182) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -2e-58) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= -5e-284) {
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	} else if ((V * l) <= 1e-299) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if ((V * l) <= 2e+302) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} 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) :: tmp
    if ((v * l) <= (-2d+182)) then
        tmp = c0_m * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-2d-58)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= (-5d-284)) then
        tmp = c0_m / (sqrt(l) * sqrt((v / a)))
    else if ((v * l) <= 1d-299) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if ((v * l) <= 2d+302) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    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 tmp;
	if ((V * l) <= -2e+182) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -2e-58) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= -5e-284) {
		tmp = c0_m / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else if ((V * l) <= 1e-299) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if ((V * l) <= 2e+302) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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):
	tmp = 0
	if (V * l) <= -2e+182:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -2e-58:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= -5e-284:
		tmp = c0_m / (math.sqrt(l) * math.sqrt((V / A)))
	elif (V * l) <= 1e-299:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif (V * l) <= 2e+302:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	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)
	tmp = 0.0
	if (Float64(V * l) <= -2e+182)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -2e-58)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= -5e-284)
		tmp = Float64(c0_m / Float64(sqrt(l) * sqrt(Float64(V / A))));
	elseif (Float64(V * l) <= 1e-299)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (Float64(V * l) <= 2e+302)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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)
	tmp = 0.0;
	if ((V * l) <= -2e+182)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -2e-58)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= -5e-284)
		tmp = c0_m / (sqrt(l) * sqrt((V / A)));
	elseif ((V * l) <= 1e-299)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif ((V * l) <= 2e+302)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	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_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -2e+182], N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-58], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-284], N[(c0$95$m / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-299], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+302], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $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 6 regimes

2. if (*.f64 V l) < -2.0000000000000001e182

   1. Initial program 57.6%

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

      1. associate-/r* 63.7%

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

      2. sqrt-div 59.3%

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

      3. associate-*r/ 59.3%

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

   4. Applied egg-rr 59.3%

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

      1. associate-/l* 59.3%

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

   6. Simplified 59.3%

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

   if -2.0000000000000001e182 < (*.f64 V l) < -2.0000000000000001e-58

   1. Initial program 94.3%

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

   if -2.0000000000000001e-58 < (*.f64 V l) < -4.99999999999999973e-284

   1. Initial program 76.9%

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

      1. *-un-lft-identity 76.9%

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

      2. times-frac 74.2%

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

   4. Applied egg-rr 74.2%

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

      1. frac-times 76.9%

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

      2. *-un-lft-identity 76.9%

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

      3. sqrt-undiv 0.0%

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

      4. clear-num 0.0%

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

      5. un-div-inv 0.0%

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

      6. sqrt-undiv 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. associate-*r/ 74.3%

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

   8. Simplified 74.3%

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

      1. associate-*r/ 77.2%

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

      2. *-un-lft-identity 77.2%

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

      3. times-frac 74.3%

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

      4. associate-/r/ 74.3%

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

      5. clear-num 74.3%

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

   10. Applied egg-rr 74.3%

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

       1. associate-/r/ 71.7%

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

       2. sqrt-prod 43.6%

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

   12. Applied egg-rr 43.6%

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

   if -4.99999999999999973e-284 < (*.f64 V l) < 9.99999999999999992e-300

   1. Initial program 49.8%

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

   3. Taylor expanded in c0 around 0 49.8%

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

      1. *-commutative 49.8%

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

      2. associate-/r* 79.6%

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

   5. Simplified 79.6%

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

   if 9.99999999999999992e-300 < (*.f64 V l) < 2.0000000000000002e302

   1. Initial program 81.2%

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

      1. sqrt-div 99.4%

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

      2. associate-*r/ 98.0%

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

   4. Applied egg-rr 98.0%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   if 2.0000000000000002e302 < (*.f64 V l)

   1. Initial program 38.6%

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

      1. add-sqr-sqrt 38.6%

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

      2. sqrt-unprod 38.6%

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

      3. *-commutative 38.6%

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

      4. *-commutative 38.6%

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

      5. swap-sqr 37.4%

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

      6. add-sqr-sqrt 37.4%

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

      7. pow2 37.4%

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

   4. Applied egg-rr 37.4%

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

      1. associate-*l/ 37.1%

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

      2. *-commutative 37.1%

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

      3. times-frac 42.3%

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

   6. Simplified 42.3%

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

      1. unpow2 42.3%

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

      2. associate-/l* 47.1%

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

   8. Applied egg-rr 47.1%

      \[\leadsto \sqrt{\frac{A}{\ell} \cdot \color{blue}{\left(c0 \cdot \frac{c0}{V}\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+182}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-58}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-299}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+302}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+182}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -2 \cdot 10^{-133}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-318}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+302}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) -2e+182)
    (* c0_m (/ (sqrt (/ A V)) (sqrt l)))
    (if (<= (* V l) -2e-133)
      (* c0_m (sqrt (/ A (* V l))))
      (if (<= (* V l) 1e-318)
        (/ c0_m (sqrt (/ V (/ A l))))
        (if (<= (* V l) 2e+302)
          (* c0_m (/ (sqrt A) (sqrt (* V l))))
          (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 tmp;
	if ((V * l) <= -2e+182) {
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	} else if ((V * l) <= -2e-133) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 1e-318) {
		tmp = c0_m / sqrt((V / (A / l)));
	} else if ((V * l) <= 2e+302) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} 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) :: tmp
    if ((v * l) <= (-2d+182)) then
        tmp = c0_m * (sqrt((a / v)) / sqrt(l))
    else if ((v * l) <= (-2d-133)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 1d-318) then
        tmp = c0_m / sqrt((v / (a / l)))
    else if ((v * l) <= 2d+302) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    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 tmp;
	if ((V * l) <= -2e+182) {
		tmp = c0_m * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else if ((V * l) <= -2e-133) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 1e-318) {
		tmp = c0_m / Math.sqrt((V / (A / l)));
	} else if ((V * l) <= 2e+302) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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):
	tmp = 0
	if (V * l) <= -2e+182:
		tmp = c0_m * (math.sqrt((A / V)) / math.sqrt(l))
	elif (V * l) <= -2e-133:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 1e-318:
		tmp = c0_m / math.sqrt((V / (A / l)))
	elif (V * l) <= 2e+302:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	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)
	tmp = 0.0
	if (Float64(V * l) <= -2e+182)
		tmp = Float64(c0_m * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	elseif (Float64(V * l) <= -2e-133)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 1e-318)
		tmp = Float64(c0_m / sqrt(Float64(V / Float64(A / l))));
	elseif (Float64(V * l) <= 2e+302)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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)
	tmp = 0.0;
	if ((V * l) <= -2e+182)
		tmp = c0_m * (sqrt((A / V)) / sqrt(l));
	elseif ((V * l) <= -2e-133)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 1e-318)
		tmp = c0_m / sqrt((V / (A / l)));
	elseif ((V * l) <= 2e+302)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	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_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -2e+182], N[(c0$95$m * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -2e-133], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-318], N[(c0$95$m / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+302], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $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 5 regimes

2. if (*.f64 V l) < -2.0000000000000001e182

   1. Initial program 57.6%

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

      1. associate-/r* 63.7%

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

      2. sqrt-div 59.3%

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

      3. associate-*r/ 59.3%

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

   4. Applied egg-rr 59.3%

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

      1. associate-/l* 59.3%

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

   6. Simplified 59.3%

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

   if -2.0000000000000001e182 < (*.f64 V l) < -2.0000000000000001e-133

   1. Initial program 92.5%

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

   if -2.0000000000000001e-133 < (*.f64 V l) < 9.9999875e-319

   1. Initial program 53.1%

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

      1. *-un-lft-identity 53.1%

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

      2. times-frac 74.3%

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

   4. Applied egg-rr 74.3%

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

      1. frac-times 53.1%

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

      2. *-un-lft-identity 53.1%

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

      3. sqrt-undiv 9.7%

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

      4. clear-num 9.7%

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

      5. un-div-inv 9.7%

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

      6. sqrt-undiv 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-*r/ 74.3%

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

   8. Simplified 74.3%

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

      1. associate-*r/ 53.1%

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

      2. *-un-lft-identity 53.1%

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

      3. times-frac 74.3%

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

      4. associate-/r/ 74.3%

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

      5. clear-num 74.4%

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

   10. Applied egg-rr 74.4%

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

   if 9.9999875e-319 < (*.f64 V l) < 2.0000000000000002e302

   1. Initial program 81.6%

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

      1. sqrt-div 99.4%

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

      2. associate-*r/ 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   if 2.0000000000000002e302 < (*.f64 V l)

   1. Initial program 38.6%

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

      1. add-sqr-sqrt 38.6%

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

      2. sqrt-unprod 38.6%

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

      3. *-commutative 38.6%

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

      4. *-commutative 38.6%

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

      5. swap-sqr 37.4%

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

      6. add-sqr-sqrt 37.4%

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

      7. pow2 37.4%

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

   4. Applied egg-rr 37.4%

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

      1. associate-*l/ 37.1%

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

      2. *-commutative 37.1%

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

      3. times-frac 42.3%

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

   6. Simplified 42.3%

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

      1. unpow2 42.3%

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

      2. associate-/l* 47.1%

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

   8. Applied egg-rr 47.1%

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

Alternative 7: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{-133}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 10^{-318}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{+302}:\\ \;\;\;\;c0\_m \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (*
  c0_s
  (if (<= (* V l) -2e-133)
    (* c0_m (sqrt (/ A (* V l))))
    (if (<= (* V l) 1e-318)
      (/ c0_m (sqrt (/ V (/ A l))))
      (if (<= (* V l) 2e+302)
        (* c0_m (/ (sqrt A) (sqrt (* V l))))
        (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 tmp;
	if ((V * l) <= -2e-133) {
		tmp = c0_m * sqrt((A / (V * l)));
	} else if ((V * l) <= 1e-318) {
		tmp = c0_m / sqrt((V / (A / l)));
	} else if ((V * l) <= 2e+302) {
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	} 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) :: tmp
    if ((v * l) <= (-2d-133)) then
        tmp = c0_m * sqrt((a / (v * l)))
    else if ((v * l) <= 1d-318) then
        tmp = c0_m / sqrt((v / (a / l)))
    else if ((v * l) <= 2d+302) then
        tmp = c0_m * (sqrt(a) / sqrt((v * l)))
    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 tmp;
	if ((V * l) <= -2e-133) {
		tmp = c0_m * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 1e-318) {
		tmp = c0_m / Math.sqrt((V / (A / l)));
	} else if ((V * l) <= 2e+302) {
		tmp = c0_m * (Math.sqrt(A) / Math.sqrt((V * l)));
	} 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):
	tmp = 0
	if (V * l) <= -2e-133:
		tmp = c0_m * math.sqrt((A / (V * l)))
	elif (V * l) <= 1e-318:
		tmp = c0_m / math.sqrt((V / (A / l)))
	elif (V * l) <= 2e+302:
		tmp = c0_m * (math.sqrt(A) / math.sqrt((V * l)))
	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)
	tmp = 0.0
	if (Float64(V * l) <= -2e-133)
		tmp = Float64(c0_m * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 1e-318)
		tmp = Float64(c0_m / sqrt(Float64(V / Float64(A / l))));
	elseif (Float64(V * l) <= 2e+302)
		tmp = Float64(c0_m * Float64(sqrt(A) / sqrt(Float64(V * l))));
	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)
	tmp = 0.0;
	if ((V * l) <= -2e-133)
		tmp = c0_m * sqrt((A / (V * l)));
	elseif ((V * l) <= 1e-318)
		tmp = c0_m / sqrt((V / (A / l)));
	elseif ((V * l) <= 2e+302)
		tmp = c0_m * (sqrt(A) / sqrt((V * l)));
	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_] := N[(c0$95$s * If[LessEqual[N[(V * l), $MachinePrecision], -2e-133], N[(c0$95$m * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 1e-318], N[(c0$95$m / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e+302], N[(c0$95$m * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $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 4 regimes

2. if (*.f64 V l) < -2.0000000000000001e-133

   1. Initial program 79.8%

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

   if -2.0000000000000001e-133 < (*.f64 V l) < 9.9999875e-319

   1. Initial program 53.1%

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

      1. *-un-lft-identity 53.1%

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

      2. times-frac 74.3%

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

   4. Applied egg-rr 74.3%

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

      1. frac-times 53.1%

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

      2. *-un-lft-identity 53.1%

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

      3. sqrt-undiv 9.7%

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

      4. clear-num 9.7%

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

      5. un-div-inv 9.7%

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

      6. sqrt-undiv 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-*r/ 74.3%

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

   8. Simplified 74.3%

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

      1. associate-*r/ 53.1%

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

      2. *-un-lft-identity 53.1%

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

      3. times-frac 74.3%

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

      4. associate-/r/ 74.3%

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

      5. clear-num 74.4%

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

   10. Applied egg-rr 74.4%

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

   if 9.9999875e-319 < (*.f64 V l) < 2.0000000000000002e302

   1. Initial program 81.6%

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

      1. sqrt-div 99.4%

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

      2. associate-*r/ 97.1%

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

   4. Applied egg-rr 97.1%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   if 2.0000000000000002e302 < (*.f64 V l)

   1. Initial program 38.6%

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

      1. add-sqr-sqrt 38.6%

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

      2. sqrt-unprod 38.6%

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

      3. *-commutative 38.6%

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

      4. *-commutative 38.6%

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

      5. swap-sqr 37.4%

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

      6. add-sqr-sqrt 37.4%

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

      7. pow2 37.4%

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

   4. Applied egg-rr 37.4%

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

      1. associate-*l/ 37.1%

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

      2. *-commutative 37.1%

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

      3. times-frac 42.3%

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

   6. Simplified 42.3%

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

      1. unpow2 42.3%

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

      2. associate-/l* 47.1%

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

   8. Applied egg-rr 47.1%

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

Alternative 8: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0\_m \cdot \frac{c0\_m}{V}\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;c0\_m \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 2e-312)
      (sqrt (* (/ A l) (* c0_m (/ c0_m V))))
      (if (<= t_0 5e+296)
        (* c0_m (pow (/ (* V l) A) -0.5))
        (* c0_m (pow (* V (/ l A)) -0.5)))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 2e-312) {
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	} else if (t_0 <= 5e+296) {
		tmp = c0_m * pow(((V * l) / A), -0.5);
	} else {
		tmp = c0_m * pow((V * (l / A)), -0.5);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 2d-312) then
        tmp = sqrt(((a / l) * (c0_m * (c0_m / v))))
    else if (t_0 <= 5d+296) then
        tmp = c0_m * (((v * l) / a) ** (-0.5d0))
    else
        tmp = c0_m * ((v * (l / a)) ** (-0.5d0))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 2e-312) {
		tmp = Math.sqrt(((A / l) * (c0_m * (c0_m / V))));
	} else if (t_0 <= 5e+296) {
		tmp = c0_m * Math.pow(((V * l) / A), -0.5);
	} else {
		tmp = c0_m * Math.pow((V * (l / A)), -0.5);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 2e-312:
		tmp = math.sqrt(((A / l) * (c0_m * (c0_m / V))))
	elif t_0 <= 5e+296:
		tmp = c0_m * math.pow(((V * l) / A), -0.5)
	else:
		tmp = c0_m * math.pow((V * (l / A)), -0.5)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 2e-312)
		tmp = sqrt(Float64(Float64(A / l) * Float64(c0_m * Float64(c0_m / V))));
	elseif (t_0 <= 5e+296)
		tmp = Float64(c0_m * (Float64(Float64(V * l) / A) ^ -0.5));
	else
		tmp = Float64(c0_m * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 2e-312)
		tmp = sqrt(((A / l) * (c0_m * (c0_m / V))));
	elseif (t_0 <= 5e+296)
		tmp = c0_m * (((V * l) / A) ^ -0.5);
	else
		tmp = c0_m * ((V * (l / A)) ^ -0.5);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 2e-312], N[Sqrt[N[(N[(A / l), $MachinePrecision] * N[(c0$95$m * N[(c0$95$m / V), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+296], N[(c0$95$m * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 2.0000000000019e-312

   1. Initial program 41.9%

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

      1. add-sqr-sqrt 40.5%

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

      2. sqrt-unprod 40.6%

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

      3. *-commutative 40.6%

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

      4. *-commutative 40.6%

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

      5. swap-sqr 39.4%

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

      6. add-sqr-sqrt 39.4%

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

      7. pow2 39.4%

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

   4. Applied egg-rr 39.4%

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

      1. associate-*l/ 39.8%

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

      2. *-commutative 39.8%

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

      3. times-frac 43.7%

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

   6. Simplified 43.7%

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

      1. unpow2 43.7%

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

      2. associate-/l* 52.3%

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

   8. Applied egg-rr 52.3%

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

   if 2.0000000000019e-312 < (/.f64 A (*.f64 V l)) < 5.0000000000000001e296

   1. Initial program 99.2%

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

   3. Taylor expanded in c0 around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-/r* 88.3%

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

   5. Simplified 88.3%

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

      1. clear-num 88.2%

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

      2. associate-/r/ 88.3%

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

   7. Applied egg-rr 88.3%

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

      1. clear-num 88.3%

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

      2. sqrt-div 88.3%

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

      3. metadata-eval 88.3%

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

      4. div-inv 88.1%

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

      5. associate-*l/ 88.2%

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

      6. *-un-lft-identity 88.2%

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

      7. clear-num 88.8%

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

      8. pow1/2 88.8%

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

      9. pow-flip 88.9%

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

      10. clear-num 88.2%

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

      11. *-un-lft-identity 88.2%

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

      12. associate-*l/ 88.2%

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

      13. div-inv 88.3%

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

      14. *-un-lft-identity 88.3%

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

      15. times-frac 88.0%

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

      16. clear-num 88.1%

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

      17. /-rgt-identity 88.1%

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

      18. metadata-eval 88.1%

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

   9. Applied egg-rr 88.1%

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

       1. associate-*r/ 99.2%

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

       2. associate-*l/ 88.9%

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

       3. *-commutative 88.9%

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

   11. Simplified 88.9%

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

       1. *-commutative 88.9%

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

       2. associate-*l/ 99.2%

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

   13. Applied egg-rr 99.2%

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

   if 5.0000000000000001e296 < (/.f64 A (*.f64 V l))

   1. Initial program 36.4%

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

   3. Taylor expanded in c0 around 0 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-/r* 54.0%

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

   5. Simplified 54.0%

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

      1. clear-num 53.9%

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

      2. associate-/r/ 53.9%

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

   7. Applied egg-rr 53.9%

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

      1. clear-num 53.9%

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

      2. sqrt-div 55.7%

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

      3. metadata-eval 55.7%

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

      4. div-inv 55.8%

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

      5. associate-*l/ 55.8%

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

      6. *-un-lft-identity 55.8%

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

      7. clear-num 55.8%

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

      8. pow1/2 55.8%

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

      9. pow-flip 55.8%

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

      10. clear-num 55.9%

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

      11. *-un-lft-identity 55.9%

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

      12. associate-*l/ 55.8%

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

      13. div-inv 55.8%

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

      14. *-un-lft-identity 55.8%

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

      15. times-frac 54.6%

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

      16. clear-num 54.7%

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

      17. /-rgt-identity 54.7%

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

      18. metadata-eval 54.7%

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

   9. Applied egg-rr 54.7%

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

       1. associate-*r/ 39.1%

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

       2. associate-*l/ 55.8%

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

       3. *-commutative 55.8%

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

   11. Simplified 55.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-312}:\\ \;\;\;\;\sqrt{\frac{A}{\ell} \cdot \left(c0 \cdot \frac{c0}{V}\right)}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-297}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;c0\_m \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 2e-297)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 5e+296)
        (* c0_m (pow (/ (* V l) A) -0.5))
        (* c0_m (pow (* V (/ l A)) -0.5)))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 2e-297) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+296) {
		tmp = c0_m * pow(((V * l) / A), -0.5);
	} else {
		tmp = c0_m * pow((V * (l / A)), -0.5);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 2d-297) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 5d+296) then
        tmp = c0_m * (((v * l) / a) ** (-0.5d0))
    else
        tmp = c0_m * ((v * (l / a)) ** (-0.5d0))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 2e-297) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+296) {
		tmp = c0_m * Math.pow(((V * l) / A), -0.5);
	} else {
		tmp = c0_m * Math.pow((V * (l / A)), -0.5);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 2e-297:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+296:
		tmp = c0_m * math.pow(((V * l) / A), -0.5)
	else:
		tmp = c0_m * math.pow((V * (l / A)), -0.5)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 2e-297)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+296)
		tmp = Float64(c0_m * (Float64(Float64(V * l) / A) ^ -0.5));
	else
		tmp = Float64(c0_m * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 2e-297)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+296)
		tmp = c0_m * (((V * l) / A) ^ -0.5);
	else
		tmp = c0_m * ((V * (l / A)) ^ -0.5);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 2e-297], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+296], N[(c0$95$m * N[Power[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 2.00000000000000008e-297

   1. Initial program 44.7%

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

   3. Taylor expanded in c0 around 0 44.7%

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

      1. *-commutative 44.7%

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

      2. associate-/r* 55.3%

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

   5. Simplified 55.3%

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

   if 2.00000000000000008e-297 < (/.f64 A (*.f64 V l)) < 5.0000000000000001e296

   1. Initial program 99.2%

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

   3. Taylor expanded in c0 around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-/r* 89.5%

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

   5. Simplified 89.5%

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

      1. clear-num 89.3%

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

      2. associate-/r/ 89.5%

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

   7. Applied egg-rr 89.5%

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

      1. clear-num 89.4%

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

      2. sqrt-div 89.5%

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

      3. metadata-eval 89.5%

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

      4. div-inv 89.3%

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

      5. associate-*l/ 89.3%

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

      6. *-un-lft-identity 89.3%

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

      7. clear-num 90.0%

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

      8. pow1/2 90.0%

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

      9. pow-flip 90.1%

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

      10. clear-num 89.4%

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

      11. *-un-lft-identity 89.4%

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

      12. associate-*l/ 89.3%

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

      13. div-inv 89.5%

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

      14. *-un-lft-identity 89.5%

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

      15. times-frac 87.7%

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

      16. clear-num 87.8%

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

      17. /-rgt-identity 87.8%

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

      18. metadata-eval 87.8%

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

   9. Applied egg-rr 87.8%

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

       1. associate-*r/ 99.2%

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

       2. associate-*l/ 90.1%

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

       3. *-commutative 90.1%

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

   11. Simplified 90.1%

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

       1. *-commutative 90.1%

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

       2. associate-*l/ 99.2%

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

   13. Applied egg-rr 99.2%

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

   if 5.0000000000000001e296 < (/.f64 A (*.f64 V l))

   1. Initial program 36.4%

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

   3. Taylor expanded in c0 around 0 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-/r* 54.0%

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

   5. Simplified 54.0%

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

      1. clear-num 53.9%

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

      2. associate-/r/ 53.9%

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

   7. Applied egg-rr 53.9%

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

      1. clear-num 53.9%

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

      2. sqrt-div 55.7%

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

      3. metadata-eval 55.7%

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

      4. div-inv 55.8%

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

      5. associate-*l/ 55.8%

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

      6. *-un-lft-identity 55.8%

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

      7. clear-num 55.8%

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

      8. pow1/2 55.8%

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

      9. pow-flip 55.8%

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

      10. clear-num 55.9%

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

      11. *-un-lft-identity 55.9%

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

      12. associate-*l/ 55.8%

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

      13. div-inv 55.8%

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

      14. *-un-lft-identity 55.8%

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

      15. times-frac 54.6%

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

      16. clear-num 54.7%

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

      17. /-rgt-identity 54.7%

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

      18. metadata-eval 54.7%

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

   9. Applied egg-rr 54.7%

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

       1. associate-*r/ 39.1%

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

       2. associate-*l/ 55.8%

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

       3. *-commutative 55.8%

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

   11. Simplified 55.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-297}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+296}:\\ \;\;\;\;c0 \cdot {\left(\frac{V \cdot \ell}{A}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-297}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (<= t_0 5e-297)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 5e+296)
        (/ c0_m (sqrt (/ (* V l) A)))
        (* c0_m (pow (* V (/ l A)) -0.5)))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-297) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+296) {
		tmp = c0_m / sqrt(((V * l) / A));
	} else {
		tmp = c0_m * pow((V * (l / A)), -0.5);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-297) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 5d+296) then
        tmp = c0_m / sqrt(((v * l) / a))
    else
        tmp = c0_m * ((v * (l / a)) ** (-0.5d0))
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-297) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+296) {
		tmp = c0_m / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0_m * Math.pow((V * (l / A)), -0.5);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-297:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+296:
		tmp = c0_m / math.sqrt(((V * l) / A))
	else:
		tmp = c0_m * math.pow((V * (l / A)), -0.5)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-297)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+296)
		tmp = Float64(c0_m / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0_m * (Float64(V * Float64(l / A)) ^ -0.5));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-297)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+296)
		tmp = c0_m / sqrt(((V * l) / A));
	else
		tmp = c0_m * ((V * (l / A)) ^ -0.5);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 5e-297], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+296], N[(c0$95$m / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Power[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 5e-297

   1. Initial program 45.6%

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

   3. Taylor expanded in c0 around 0 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-/r* 56.0%

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

   5. Simplified 56.0%

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

   if 5e-297 < (/.f64 A (*.f64 V l)) < 5.0000000000000001e296

   1. Initial program 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. times-frac 89.5%

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

   4. Applied egg-rr 89.5%

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

      1. frac-times 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. sqrt-undiv 47.8%

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

      4. clear-num 47.8%

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

      5. un-div-inv 47.8%

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

      6. sqrt-undiv 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 5.0000000000000001e296 < (/.f64 A (*.f64 V l))

   1. Initial program 36.4%

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

   3. Taylor expanded in c0 around 0 36.4%

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

      1. *-commutative 36.4%

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

      2. associate-/r* 54.0%

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

   5. Simplified 54.0%

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

      1. clear-num 53.9%

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

      2. associate-/r/ 53.9%

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

   7. Applied egg-rr 53.9%

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

      1. clear-num 53.9%

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

      2. sqrt-div 55.7%

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

      3. metadata-eval 55.7%

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

      4. div-inv 55.8%

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

      5. associate-*l/ 55.8%

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

      6. *-un-lft-identity 55.8%

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

      7. clear-num 55.8%

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

      8. pow1/2 55.8%

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

      9. pow-flip 55.8%

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

      10. clear-num 55.9%

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

      11. *-un-lft-identity 55.9%

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

      12. associate-*l/ 55.8%

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

      13. div-inv 55.8%

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

      14. *-un-lft-identity 55.8%

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

      15. times-frac 54.6%

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

      16. clear-num 54.7%

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

      17. /-rgt-identity 54.7%

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

      18. metadata-eval 54.7%

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

   9. Applied egg-rr 54.7%

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

       1. associate-*r/ 39.1%

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

       2. associate-*l/ 55.8%

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

       3. *-commutative 55.8%

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

   11. Simplified 55.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-297}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot {\left(V \cdot \frac{\ell}{A}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 5 \cdot 10^{+262}\right):\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0\_m \cdot \sqrt{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

c0\_m = (fabs.f64 c0)
c0\_s = (copysign.f64 #s(literal 1 binary64) c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0_s c0_m A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (*
    c0_s
    (if (or (<= t_0 0.0) (not (<= t_0 5e+262)))
      (* c0_m (sqrt (/ (/ A V) l)))
      (* c0_m (sqrt t_0))))))

C

c0\_m = fabs(c0);
c0\_s = copysign(1.0, c0);
assert(c0_m < A && A < V && V < l);
double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+262)) {
		tmp = c0_m * sqrt(((A / V) / l));
	} else {
		tmp = c0_m * sqrt(t_0);
	}
	return c0_s * tmp;
}

Fortran

c0\_m = abs(c0)
c0\_s = copysign(1.0d0, c0)
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0_s, c0_m, a, v, l)
    real(8), intent (in) :: c0_s
    real(8), intent (in) :: c0_m
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 5d+262))) then
        tmp = c0_m * sqrt(((a / v) / l))
    else
        tmp = c0_m * sqrt(t_0)
    end if
    code = c0_s * tmp
end function

Java

c0\_m = Math.abs(c0);
c0\_s = Math.copySign(1.0, c0);
assert c0_m < A && A < V && V < l;
public static double code(double c0_s, double c0_m, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+262)) {
		tmp = c0_m * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0_m * Math.sqrt(t_0);
	}
	return c0_s * tmp;
}

Python

c0\_m = math.fabs(c0)
c0\_s = math.copysign(1.0, c0)
[c0_m, A, V, l] = sort([c0_m, A, V, l])
def code(c0_s, c0_m, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+262):
		tmp = c0_m * math.sqrt(((A / V) / l))
	else:
		tmp = c0_m * math.sqrt(t_0)
	return c0_s * tmp

Julia

c0\_m = abs(c0)
c0\_s = copysign(1.0, c0)
c0_m, A, V, l = sort([c0_m, A, V, l])
function code(c0_s, c0_m, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+262))
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0_m * sqrt(t_0));
	end
	return Float64(c0_s * tmp)
end

MATLAB

c0\_m = abs(c0);
c0\_s = sign(c0) * abs(1.0);
c0_m, A, V, l = num2cell(sort([c0_m, A, V, l])){:}
function tmp_2 = code(c0_s, c0_m, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+262)))
		tmp = c0_m * sqrt(((A / V) / l));
	else
		tmp = c0_m * sqrt(t_0);
	end
	tmp_2 = c0_s * tmp;
end

Wolfram

c0\_m = N[Abs[c0], $MachinePrecision]
c0\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: c0_m, A, V, and l should be sorted in increasing order before calling this function.
code[c0$95$s_, c0$95$m_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 5e+262]], $MachinePrecision]], N[(c0$95$m * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 0.0 or 5.00000000000000008e262 < (/.f64 A (*.f64 V l))

   1. Initial program 41.0%

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

      1. associate-/r* 56.6%

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

   3. Simplified 56.6%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.00000000000000008e262

   1. Initial program 98.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+262}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} c0\_m = \left|c0\right| \\ c0\_s = \mathsf{copysign}\left(1, c0\right) \\ [c0_m, A, V, l] = \mathsf{sort}([c0_m, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ c0\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-297}:\\ \;\;\;\;c0\_m \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0\_m}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \end{array} \end{array} \]

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 5e-297)
      (* c0_m (sqrt (/ (/ A l) V)))
      (if (<= t_0 5e+296)
        (/ c0_m (sqrt (/ (* V l) A)))
        (/ c0_m (sqrt (/ V (/ A 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 t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-297) {
		tmp = c0_m * sqrt(((A / l) / V));
	} else if (t_0 <= 5e+296) {
		tmp = c0_m / sqrt(((V * l) / A));
	} else {
		tmp = c0_m / sqrt((V / (A / 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) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-297) then
        tmp = c0_m * sqrt(((a / l) / v))
    else if (t_0 <= 5d+296) then
        tmp = c0_m / sqrt(((v * l) / a))
    else
        tmp = c0_m / sqrt((v / (a / 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 t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-297) {
		tmp = c0_m * Math.sqrt(((A / l) / V));
	} else if (t_0 <= 5e+296) {
		tmp = c0_m / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0_m / Math.sqrt((V / (A / 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):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-297:
		tmp = c0_m * math.sqrt(((A / l) / V))
	elif t_0 <= 5e+296:
		tmp = c0_m / math.sqrt(((V * l) / A))
	else:
		tmp = c0_m / math.sqrt((V / (A / 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)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-297)
		tmp = Float64(c0_m * sqrt(Float64(Float64(A / l) / V)));
	elseif (t_0 <= 5e+296)
		tmp = Float64(c0_m / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0_m / sqrt(Float64(V / Float64(A / 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)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-297)
		tmp = c0_m * sqrt(((A / l) / V));
	elseif (t_0 <= 5e+296)
		tmp = c0_m / sqrt(((V * l) / A));
	else
		tmp = c0_m / sqrt((V / (A / 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_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, N[(c0$95$s * If[LessEqual[t$95$0, 5e-297], N[(c0$95$m * N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+296], N[(c0$95$m / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0$95$m / N[Sqrt[N[(V / N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 5e-297

   1. Initial program 45.6%

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

   3. Taylor expanded in c0 around 0 45.6%

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

      1. *-commutative 45.6%

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

      2. associate-/r* 56.0%

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

   5. Simplified 56.0%

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

   if 5e-297 < (/.f64 A (*.f64 V l)) < 5.0000000000000001e296

   1. Initial program 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. times-frac 89.5%

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

   4. Applied egg-rr 89.5%

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

      1. frac-times 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. sqrt-undiv 47.8%

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

      4. clear-num 47.8%

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

      5. un-div-inv 47.8%

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

      6. sqrt-undiv 99.2%

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

   6. Applied egg-rr 99.2%

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

   if 5.0000000000000001e296 < (/.f64 A (*.f64 V l))

   1. Initial program 36.4%

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

      1. *-un-lft-identity 36.4%

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

      2. times-frac 53.9%

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

   4. Applied egg-rr 53.9%

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

      1. frac-times 36.4%

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

      2. *-un-lft-identity 36.4%

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

      3. sqrt-undiv 35.8%

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

      4. clear-num 35.8%

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

      5. un-div-inv 35.8%

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

      6. sqrt-undiv 39.1%

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

   6. Applied egg-rr 39.1%

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

      1. associate-*r/ 55.8%

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

   8. Simplified 55.8%

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

      1. associate-*r/ 39.1%

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

      2. *-un-lft-identity 39.1%

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

      3. times-frac 55.8%

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

      4. associate-/r/ 55.8%

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

      5. clear-num 55.8%

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

   10. Applied egg-rr 55.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-297}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V}{\frac{A}{\ell}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.2% 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 71.5%

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

Reproduce

herbie shell --seed 2024146 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))