Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.3% → 91.1%

Time: 17.2s

Alternatives: 11

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: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{-A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -2e-310)
   (/ c0 (* (/ (sqrt (- V)) (sqrt (- A))) (sqrt l)))
   (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 / ((sqrt(-V) / sqrt(-A)) * sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if (a <= (-2d-310)) then
        tmp = c0 / ((sqrt(-v) / sqrt(-a)) * sqrt(l))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -2e-310) {
		tmp = c0 / ((Math.sqrt(-V) / Math.sqrt(-A)) * Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -2e-310:
		tmp = c0 / ((math.sqrt(-V) / math.sqrt(-A)) * math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -2e-310)
		tmp = Float64(c0 / Float64(Float64(sqrt(Float64(-V)) / sqrt(Float64(-A))) * sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (A <= -2e-310)
		tmp = c0 / ((sqrt(-V) / sqrt(-A)) * sqrt(l));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -2e-310], N[(c0 / N[(N[(N[Sqrt[(-V)], $MachinePrecision] / N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -1.999999999999994e-310

   1. Initial program 71.9%

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

      1. *-un-lft-identity 71.9%

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

      2. times-frac 72.7%

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

   4. Applied egg-rr 72.7%

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

      1. frac-times 71.9%

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

      2. *-un-lft-identity 71.9%

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

      3. *-un-lft-identity 71.9%

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

      4. *-commutative 71.9%

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

      5. times-frac 72.7%

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

      6. sqrt-unprod 41.2%

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

      7. *-commutative 41.2%

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

      8. expm1-log1p-u 32.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)\right)} \]

      9. expm1-udef 14.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)} - 1} \]

   6. Applied egg-rr 28.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 59.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

      2. expm1-log1p 72.8%

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

   8. Simplified 72.8%

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

      1. *-commutative 72.8%

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

      2. sqrt-prod 42.9%

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

   10. Applied egg-rr 42.9%

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

       1. frac-2neg 42.9%

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

       2. sqrt-div 49.0%

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

   12. Applied egg-rr 49.0%

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

   if -1.999999999999994e-310 < A

   1. Initial program 80.0%

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

      1. div-inv 80.0%

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

      2. sqrt-prod 87.8%

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

      3. associate-/r* 87.9%

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

   4. Applied egg-rr 87.9%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{-V}}{\sqrt{-A}} \cdot \sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 10^{+251}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (or (<= t_0 0.0) (not (<= t_0 1e+251)))
     (* c0 (sqrt (/ (/ A V) l)))
     t_0)))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1e+251)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if ((t_0 <= 0.0d0) .or. (.not. (t_0 <= 1d+251))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 1e+251)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 1e+251):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = t_0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 1e+251))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 1e+251)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1e+251]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 0.0 or 1e251 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 66.5%

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

      1. *-commutative 66.5%

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

      2. associate-/l/ 73.1%

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

   3. Simplified 73.1%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1e251

   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 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0 \lor \neg \left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+251}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+211}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 2e+211) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 2e+211) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 2d+211) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 2e+211) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 2e+211:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 2e+211)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 2e+211)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+211], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $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 68.5%

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

      1. *-commutative 68.5%

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

      2. associate-/l/ 75.4%

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

   3. Simplified 75.4%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.9999999999999999e211

   1. Initial program 98.4%

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

   if 1.9999999999999999e211 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. times-frac 66.6%

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

   4. Applied egg-rr 66.6%

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

      1. frac-times 61.9%

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

      2. *-un-lft-identity 61.9%

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

      3. *-un-lft-identity 61.9%

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

      4. *-commutative 61.9%

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

      5. times-frac 66.7%

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

      6. sqrt-unprod 35.2%

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

      7. *-commutative 35.2%

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

      8. expm1-log1p-u 33.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)\right)} \]

      9. expm1-udef 30.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)} - 1} \]

   6. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 63.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

      2. expm1-log1p 66.6%

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

      3. *-commutative 66.6%

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

      4. associate-*l/ 62.7%

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

      5. associate-*r/ 67.6%

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

   8. Simplified 67.6%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+211}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 10^{-306}:\\ \;\;\;\;\frac{c0}{\frac{1}{\sqrt{\frac{\frac{A}{\ell}}{V}}}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+211}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* c0 (sqrt (/ A (* V l))))))
   (if (<= t_0 1e-306)
     (/ c0 (/ 1.0 (sqrt (/ (/ A l) V))))
     (if (<= t_0 2e+211) t_0 (/ c0 (sqrt (* V (/ l A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 * sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-306) {
		tmp = c0 / (1.0 / sqrt(((A / l) / V)));
	} else if (t_0 <= 2e+211) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c0 * sqrt((a / (v * l)))
    if (t_0 <= 1d-306) then
        tmp = c0 / (1.0d0 / sqrt(((a / l) / v)))
    else if (t_0 <= 2d+211) then
        tmp = t_0
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = c0 * Math.sqrt((A / (V * l)));
	double tmp;
	if (t_0 <= 1e-306) {
		tmp = c0 / (1.0 / Math.sqrt(((A / l) / V)));
	} else if (t_0 <= 2e+211) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 * math.sqrt((A / (V * l)))
	tmp = 0
	if t_0 <= 1e-306:
		tmp = c0 / (1.0 / math.sqrt(((A / l) / V)))
	elif t_0 <= 2e+211:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 * sqrt(Float64(A / Float64(V * l))))
	tmp = 0.0
	if (t_0 <= 1e-306)
		tmp = Float64(c0 / Float64(1.0 / sqrt(Float64(Float64(A / l) / V))));
	elseif (t_0 <= 2e+211)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = c0 * sqrt((A / (V * l)));
	tmp = 0.0;
	if (t_0 <= 1e-306)
		tmp = c0 / (1.0 / sqrt(((A / l) / V)));
	elseif (t_0 <= 2e+211)
		tmp = t_0;
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-306], N[(c0 / N[(1.0 / N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+211], t$95$0, N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.00000000000000003e-306

   1. Initial program 68.7%

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

      1. *-un-lft-identity 68.7%

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

      2. times-frac 72.4%

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

   4. Applied egg-rr 72.4%

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

      1. frac-times 68.7%

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

      2. *-un-lft-identity 68.7%

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

      3. *-un-lft-identity 68.7%

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

      4. *-commutative 68.7%

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

      5. times-frac 75.5%

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

      6. sqrt-unprod 47.7%

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

      7. *-commutative 47.7%

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

      8. expm1-log1p-u 27.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)\right)} \]

      9. expm1-udef 8.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)} - 1} \]

   6. Applied egg-rr 15.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 44.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

      2. expm1-log1p 74.8%

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

   8. Simplified 74.8%

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

      1. *-commutative 74.8%

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

      2. associate-*l/ 68.5%

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

      3. associate-/l* 72.3%

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

   10. Applied egg-rr 72.3%

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

       1. associate-/r/ 74.8%

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

       2. sqrt-unprod 48.3%

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

       3. *-commutative 48.3%

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

       4. sqrt-prod 74.8%

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

       5. clear-num 74.8%

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

       6. div-inv 74.8%

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

       7. clear-num 74.5%

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

       8. sqrt-div 75.5%

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

       9. metadata-eval 75.5%

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

       10. associate-/l/ 68.6%

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

       11. *-commutative 68.6%

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

   12. Applied egg-rr 68.6%

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

       1. *-commutative 68.6%

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

       2. associate-/r* 72.3%

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

   14. Simplified 72.3%

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

   if 1.00000000000000003e-306 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.9999999999999999e211

   1. Initial program 98.4%

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

   if 1.9999999999999999e211 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. times-frac 66.6%

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

   4. Applied egg-rr 66.6%

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

      1. frac-times 61.9%

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

      2. *-un-lft-identity 61.9%

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

      3. *-un-lft-identity 61.9%

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

      4. *-commutative 61.9%

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

      5. times-frac 66.7%

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

      6. sqrt-unprod 35.2%

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

      7. *-commutative 35.2%

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

      8. expm1-log1p-u 33.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)\right)} \]

      9. expm1-udef 30.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)} - 1} \]

   6. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 63.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

      2. expm1-log1p 66.6%

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

      3. *-commutative 66.6%

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

      4. associate-*l/ 62.7%

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

      5. associate-*r/ 67.6%

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

   8. Simplified 67.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{-306}:\\ \;\;\;\;\frac{c0}{\frac{1}{\sqrt{\frac{\frac{A}{\ell}}{V}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 2 \cdot 10^{+211}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-166}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(V \cdot \ell\right) \cdot \frac{1}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -2e+187)
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -1e-166)
     (/ c0 (sqrt (* (* V l) (/ 1.0 A))))
     (if (<= (* V l) 0.0)
       (/ (/ c0 (sqrt (/ V A))) (sqrt l))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+187) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -1e-166) {
		tmp = c0 / sqrt(((V * l) * (1.0 / A)));
	} else if ((V * l) <= 0.0) {
		tmp = (c0 / sqrt((V / A))) / sqrt(l);
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-2d+187)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((v * l) <= (-1d-166)) then
        tmp = c0 / sqrt(((v * l) * (1.0d0 / a)))
    else if ((v * l) <= 0.0d0) then
        tmp = (c0 / sqrt((v / a))) / sqrt(l)
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -2e+187) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -1e-166) {
		tmp = c0 / Math.sqrt(((V * l) * (1.0 / A)));
	} else if ((V * l) <= 0.0) {
		tmp = (c0 / Math.sqrt((V / A))) / Math.sqrt(l);
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -2e+187:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -1e-166:
		tmp = c0 / math.sqrt(((V * l) * (1.0 / A)))
	elif (V * l) <= 0.0:
		tmp = (c0 / math.sqrt((V / A))) / math.sqrt(l)
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -2e+187)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -1e-166)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) * Float64(1.0 / A))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(Float64(c0 / sqrt(Float64(V / A))) / sqrt(l));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -2e+187)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -1e-166)
		tmp = c0 / sqrt(((V * l) * (1.0 / A)));
	elseif ((V * l) <= 0.0)
		tmp = (c0 / sqrt((V / A))) / sqrt(l);
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -2e+187], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -1e-166], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] * N[(1.0 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(N[(c0 / N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1.99999999999999981e187

   1. Initial program 36.1%

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

      1. *-un-lft-identity 36.1%

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

      2. times-frac 63.0%

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

   4. Applied egg-rr 63.0%

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

      1. *-commutative 63.0%

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

      2. frac-times 36.1%

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

      3. *-un-lft-identity 36.1%

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

      4. sqrt-div 0.0%

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

      5. associate-*l/ 0.0%

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

      6. sqrt-prod 0.0%

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

      7. associate-/r* 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 41.1%

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

   6. Applied egg-rr 41.1%

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

      1. *-commutative 41.1%

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

      2. associate-*r/ 44.0%

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

   8. Simplified 44.0%

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

   if -1.99999999999999981e187 < (*.f64 V l) < -1.00000000000000004e-166

   1. Initial program 95.9%

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

      1. *-un-lft-identity 95.9%

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

      2. times-frac 82.5%

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

   4. Applied egg-rr 82.5%

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

      1. frac-times 95.9%

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

      2. *-un-lft-identity 95.9%

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

      3. *-un-lft-identity 95.9%

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

      4. *-commutative 95.9%

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

      5. times-frac 82.4%

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

      6. sqrt-unprod 41.1%

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

      7. *-commutative 41.1%

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

      8. expm1-log1p-u 35.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)\right)} \]

      9. expm1-udef 13.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)} - 1} \]

   6. Applied egg-rr 26.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 71.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

      2. expm1-log1p 84.4%

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

   8. Simplified 84.4%

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

      1. associate-*r/ 96.4%

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

      2. *-commutative 96.4%

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

      3. clear-num 95.9%

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

   10. Applied egg-rr 95.9%

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

       1. associate-/r/ 96.5%

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

   12. Simplified 96.5%

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

   if -1.00000000000000004e-166 < (*.f64 V l) < -0.0

   1. Initial program 54.0%

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

      1. *-commutative 54.0%

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

      2. associate-/r* 65.8%

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

      3. sqrt-div 44.0%

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

      4. associate-*l/ 44.0%

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

   4. Applied egg-rr 44.0%

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

      1. *-commutative 44.0%

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

      2. clear-num 44.0%

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

      3. sqrt-div 45.9%

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

      4. metadata-eval 45.9%

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

      5. un-div-inv 45.8%

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

   6. Applied egg-rr 45.8%

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

   if -0.0 < (*.f64 V l)

   1. Initial program 83.0%

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

      1. div-inv 83.0%

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

      2. sqrt-prod 91.9%

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

      3. associate-/r* 92.0%

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

   4. Applied egg-rr 92.0%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -2 \cdot 10^{+187}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-166}:\\ \;\;\;\;\frac{c0}{\sqrt{\left(V \cdot \ell\right) \cdot \frac{1}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\frac{V}{A}}}}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+270}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\frac{V}{A}}}{{\ell}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= (* V l) -1e+270)
   (* (sqrt (/ A V)) (/ c0 (sqrt l)))
   (if (<= (* V l) -5e-322)
     (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
     (if (<= (* V l) 0.0)
       (/ c0 (/ (sqrt (/ V A)) (pow l -0.5)))
       (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+270) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((V * l) <= -5e-322) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (sqrt((V / A)) / pow(l, -0.5));
	} else {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((v * l) <= (-1d+270)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((v * l) <= (-5d-322)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else if ((v * l) <= 0.0d0) then
        tmp = c0 / (sqrt((v / a)) / (l ** (-0.5d0)))
    else
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if ((V * l) <= -1e+270) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((V * l) <= -5e-322) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if ((V * l) <= 0.0) {
		tmp = c0 / (Math.sqrt((V / A)) / Math.pow(l, -0.5));
	} else {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if (V * l) <= -1e+270:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (V * l) <= -5e-322:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif (V * l) <= 0.0:
		tmp = c0 / (math.sqrt((V / A)) / math.pow(l, -0.5))
	else:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (Float64(V * l) <= -1e+270)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(V * l) <= -5e-322)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif (Float64(V * l) <= 0.0)
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) / (l ^ -0.5)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if ((V * l) <= -1e+270)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((V * l) <= -5e-322)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif ((V * l) <= 0.0)
		tmp = c0 / (sqrt((V / A)) / (l ^ -0.5));
	else
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[N[(V * l), $MachinePrecision], -1e+270], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-322], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] / N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -1e270

   1. Initial program 33.5%

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

      1. *-un-lft-identity 33.5%

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

      2. times-frac 67.9%

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

   4. Applied egg-rr 67.9%

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

      1. *-commutative 67.9%

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

      2. frac-times 33.5%

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

      3. *-un-lft-identity 33.5%

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

      4. sqrt-div 0.0%

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

      5. associate-*l/ 0.0%

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

      6. sqrt-prod 0.0%

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

      7. associate-/r* 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r/ 0.0%

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

      10. sqrt-div 36.5%

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

   6. Applied egg-rr 36.5%

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

      1. *-commutative 36.5%

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

      2. associate-*r/ 40.2%

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

   8. Simplified 40.2%

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

   if -1e270 < (*.f64 V l) < -4.99006e-322

   1. Initial program 88.6%

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

      1. frac-2neg 88.6%

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

      2. sqrt-div 98.5%

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

      3. distribute-rgt-neg-in 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. distribute-rgt-neg-out 98.5%

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

      2. *-commutative 98.5%

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

      3. distribute-rgt-neg-in 98.5%

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

   6. Simplified 98.5%

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

   if -4.99006e-322 < (*.f64 V l) < -0.0

   1. Initial program 49.0%

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

      1. pow1/2 49.0%

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

      2. associate-/r* 64.2%

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

      3. div-inv 64.2%

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

      4. unpow-prod-down 45.9%

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

      5. pow1/2 45.9%

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

   4. Applied egg-rr 45.9%

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

      1. unpow1/2 45.9%

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

   6. Simplified 45.9%

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

      1. clear-num 45.9%

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

      2. sqrt-div 48.4%

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

      3. metadata-eval 48.4%

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

   8. Applied egg-rr 48.4%

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

      1. associate-*l/ 48.4%

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

      2. *-un-lft-identity 48.4%

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

      3. associate-*r/ 48.3%

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

      4. inv-pow 48.3%

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

      5. sqrt-pow1 48.3%

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

      6. metadata-eval 48.3%

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

   10. Applied egg-rr 48.3%

       \[\leadsto \color{blue}{\frac{c0 \cdot {\ell}^{-0.5}}{\sqrt{\frac{V}{A}}}} \]
   11. Step-by-step derivation

       1. associate-/l* 48.4%

          \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\frac{V}{A}}}{{\ell}^{-0.5}}}} \]

   12. Simplified 48.4%

       \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\frac{V}{A}}}{{\ell}^{-0.5}}}} \]

   if -0.0 < (*.f64 V l)

   1. Initial program 83.0%

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

      1. div-inv 83.0%

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

      2. sqrt-prod 91.9%

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

      3. associate-/r* 92.0%

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

   4. Applied egg-rr 92.0%

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -1 \cdot 10^{+270}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-322}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\frac{V}{A}}}{{\ell}^{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+306}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\frac{V}{A}}}{{\ell}^{-0.5}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-317)
     (* (sqrt (/ A V)) (/ c0 (sqrt l)))
     (if (<= t_0 1e+306)
       (* c0 (sqrt t_0))
       (/ c0 (/ (sqrt (/ V A)) (pow l -0.5)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if (t_0 <= 1e+306) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / (sqrt((V / A)) / pow(l, -0.5));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-317) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if (t_0 <= 1d+306) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / (sqrt((v / a)) / (l ** (-0.5d0)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if (t_0 <= 1e+306) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / (Math.sqrt((V / A)) / Math.pow(l, -0.5));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-317:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif t_0 <= 1e+306:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / (math.sqrt((V / A)) / math.pow(l, -0.5))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-317)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (t_0 <= 1e+306)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V / A)) / (l ^ -0.5)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-317)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif (t_0 <= 1e+306)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / (sqrt((V / A)) / (l ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-317], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+306], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision] / N[Power[l, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 5.00000017e-317

   1. Initial program 30.5%

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

      1. *-un-lft-identity 30.5%

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

      2. times-frac 56.2%

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

   4. Applied egg-rr 56.2%

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

      1. *-commutative 56.2%

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

      2. frac-times 30.5%

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

      3. *-un-lft-identity 30.5%

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

      4. sqrt-div 20.5%

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

      5. associate-*l/ 20.6%

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

      6. sqrt-prod 18.7%

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

      7. associate-/r* 18.7%

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

      8. *-commutative 18.7%

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

      9. associate-*r/ 18.7%

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

      10. sqrt-div 46.0%

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

   6. Applied egg-rr 46.0%

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

      1. *-commutative 46.0%

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

      2. associate-*r/ 46.0%

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

   8. Simplified 46.0%

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

   if 5.00000017e-317 < (/.f64 A (*.f64 V l)) < 1.00000000000000002e306

   1. Initial program 99.2%

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

   if 1.00000000000000002e306 < (/.f64 A (*.f64 V l))

   1. Initial program 43.3%

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

      1. pow1/2 43.3%

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

      2. associate-/r* 54.5%

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

      3. div-inv 54.4%

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

      4. unpow-prod-down 40.3%

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

      5. pow1/2 40.3%

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

   4. Applied egg-rr 40.3%

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

      1. unpow1/2 40.3%

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

   6. Simplified 40.3%

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

      1. clear-num 40.3%

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

      2. sqrt-div 42.1%

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

      3. metadata-eval 42.1%

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

   8. Applied egg-rr 42.1%

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

      1. associate-*l/ 42.1%

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

      2. *-un-lft-identity 42.1%

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

      3. associate-*r/ 42.1%

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

      4. inv-pow 42.1%

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

      5. sqrt-pow1 42.1%

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

      6. metadata-eval 42.1%

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

   10. Applied egg-rr 42.1%

       \[\leadsto \color{blue}{\frac{c0 \cdot {\ell}^{-0.5}}{\sqrt{\frac{V}{A}}}} \]
   11. Step-by-step derivation

       1. associate-/l* 42.1%

          \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\frac{V}{A}}}{{\ell}^{-0.5}}}} \]

   12. Simplified 42.1%

       \[\leadsto \color{blue}{\frac{c0}{\frac{\sqrt{\frac{V}{A}}}{{\ell}^{-0.5}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 10^{+306}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\frac{V}{A}}}{{\ell}^{-0.5}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-317} \lor \neg \left(t_0 \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (or (<= t_0 5e-317) (not (<= t_0 2e+304)))
     (* (sqrt (/ A V)) (/ c0 (sqrt l)))
     (* c0 (sqrt t_0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 5e-317) || !(t_0 <= 2e+304)) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if ((t_0 <= 5d-317) .or. (.not. (t_0 <= 2d+304))) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 5e-317) || !(t_0 <= 2e+304)) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 5e-317) or not (t_0 <= 2e+304):
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 5e-317) || !(t_0 <= 2e+304))
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 5e-317) || ~((t_0 <= 2e+304)))
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e-317], N[Not[LessEqual[t$95$0, 2e+304]], $MachinePrecision]], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 A (*.f64 V l)) < 5.00000017e-317 or 1.9999999999999999e304 < (/.f64 A (*.f64 V l))

   1. Initial program 37.6%

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

      1. *-un-lft-identity 37.6%

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

      2. times-frac 55.8%

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

   4. Applied egg-rr 55.8%

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

      1. *-commutative 55.8%

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

      2. frac-times 37.6%

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

      3. *-un-lft-identity 37.6%

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

      4. sqrt-div 25.4%

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

      5. associate-*l/ 25.4%

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

      6. sqrt-prod 21.2%

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

      7. associate-/r* 21.2%

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

      8. *-commutative 21.2%

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

      9. associate-*r/ 21.1%

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

      10. sqrt-div 42.7%

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

   6. Applied egg-rr 42.7%

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

      1. *-commutative 42.7%

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

      2. associate-*r/ 42.6%

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

   8. Simplified 42.6%

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

   if 5.00000017e-317 < (/.f64 A (*.f64 V l)) < 1.9999999999999999e304

   1. Initial program 99.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-317} \lor \neg \left(\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ t_1 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-317}:\\ \;\;\;\;t_1 \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{t_1}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))) (t_1 (sqrt (/ A V))))
   (if (<= t_0 5e-317)
     (* t_1 (/ c0 (sqrt l)))
     (if (<= t_0 2e+304) (* c0 (sqrt t_0)) (* c0 (/ t_1 (sqrt l)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double t_1 = sqrt((A / V));
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = t_1 * (c0 / sqrt(l));
	} else if (t_0 <= 2e+304) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 * (t_1 / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (v * l)
    t_1 = sqrt((a / v))
    if (t_0 <= 5d-317) then
        tmp = t_1 * (c0 / sqrt(l))
    else if (t_0 <= 2d+304) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 * (t_1 / sqrt(l))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double t_1 = Math.sqrt((A / V));
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = t_1 * (c0 / Math.sqrt(l));
	} else if (t_0 <= 2e+304) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 * (t_1 / Math.sqrt(l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	t_1 = math.sqrt((A / V))
	tmp = 0
	if t_0 <= 5e-317:
		tmp = t_1 * (c0 / math.sqrt(l))
	elif t_0 <= 2e+304:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 * (t_1 / math.sqrt(l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	t_1 = sqrt(Float64(A / V))
	tmp = 0.0
	if (t_0 <= 5e-317)
		tmp = Float64(t_1 * Float64(c0 / sqrt(l)));
	elseif (t_0 <= 2e+304)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 * Float64(t_1 / sqrt(l)));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	t_1 = sqrt((A / V));
	tmp = 0.0;
	if (t_0 <= 5e-317)
		tmp = t_1 * (c0 / sqrt(l));
	elseif (t_0 <= 2e+304)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 * (t_1 / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 5e-317], N[(t$95$1 * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+304], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(t$95$1 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 5.00000017e-317

   1. Initial program 30.5%

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

      1. *-un-lft-identity 30.5%

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

      2. times-frac 56.2%

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

   4. Applied egg-rr 56.2%

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

      1. *-commutative 56.2%

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

      2. frac-times 30.5%

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

      3. *-un-lft-identity 30.5%

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

      4. sqrt-div 20.5%

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

      5. associate-*l/ 20.6%

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

      6. sqrt-prod 18.7%

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

      7. associate-/r* 18.7%

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

      8. *-commutative 18.7%

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

      9. associate-*r/ 18.7%

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

      10. sqrt-div 46.0%

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

   6. Applied egg-rr 46.0%

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

      1. *-commutative 46.0%

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

      2. associate-*r/ 46.0%

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

   8. Simplified 46.0%

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

   if 5.00000017e-317 < (/.f64 A (*.f64 V l)) < 1.9999999999999999e304

   1. Initial program 99.2%

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

   if 1.9999999999999999e304 < (/.f64 A (*.f64 V l))

   1. Initial program 44.4%

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

      1. associate-/r* 53.5%

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

      2. sqrt-div 39.5%

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

      3. associate-*r/ 39.5%

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

   4. Applied egg-rr 39.5%

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

      1. *-commutative 39.5%

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

      2. associate-/l* 39.5%

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

      3. associate-/r/ 39.5%

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

   6. Simplified 39.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* V l))))
   (if (<= t_0 5e-317)
     (* (sqrt (/ A V)) (/ c0 (sqrt l)))
     (if (<= t_0 2e+304)
       (* c0 (sqrt t_0))
       (/ c0 (* (sqrt l) (sqrt (/ V A))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if (t_0 <= 2e+304) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (v * l)
    if (t_0 <= 5d-317) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if (t_0 <= 2d+304) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-317) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if (t_0 <= 2e+304) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-317:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif t_0 <= 2e+304:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-317)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (t_0 <= 2e+304)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-317)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif (t_0 <= 2e+304)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-317], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+304], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 5.00000017e-317

   1. Initial program 30.5%

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

      1. *-un-lft-identity 30.5%

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

      2. times-frac 56.2%

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

   4. Applied egg-rr 56.2%

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

      1. *-commutative 56.2%

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

      2. frac-times 30.5%

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

      3. *-un-lft-identity 30.5%

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

      4. sqrt-div 20.5%

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

      5. associate-*l/ 20.6%

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

      6. sqrt-prod 18.7%

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

      7. associate-/r* 18.7%

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

      8. *-commutative 18.7%

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

      9. associate-*r/ 18.7%

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

      10. sqrt-div 46.0%

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

   6. Applied egg-rr 46.0%

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

      1. *-commutative 46.0%

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

      2. associate-*r/ 46.0%

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

   8. Simplified 46.0%

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

   if 5.00000017e-317 < (/.f64 A (*.f64 V l)) < 1.9999999999999999e304

   1. Initial program 99.2%

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

   if 1.9999999999999999e304 < (/.f64 A (*.f64 V l))

   1. Initial program 44.4%

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

      1. *-un-lft-identity 44.4%

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

      2. times-frac 55.3%

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

   4. Applied egg-rr 55.3%

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

      1. frac-times 44.4%

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

      2. *-un-lft-identity 44.4%

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

      3. *-un-lft-identity 44.4%

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

      4. *-commutative 44.4%

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

      5. times-frac 53.4%

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

      6. sqrt-unprod 39.5%

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

      7. *-commutative 39.5%

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

      8. expm1-log1p-u 13.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)\right)} \]

      9. expm1-udef 9.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\right)} - 1} \]

   6. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 22.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\right)\right)} \]

      2. expm1-log1p 54.4%

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

   8. Simplified 54.4%

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

      1. *-commutative 54.4%

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

      2. sqrt-prod 41.2%

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

   10. Applied egg-rr 41.2%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-317}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{+304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell} \cdot \sqrt{\frac{V}{A}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))

C

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

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    code = c0 * sqrt((a / (v * l)))
end function

Java

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

Python

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

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	return Float64(c0 * sqrt(Float64(A / Float64(V * l))))
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp = code(c0, A, V, l)
	tmp = c0 * sqrt((A / (V * l)));
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

3. Final simplification 75.4%

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

Reproduce

herbie shell --seed 2024024 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))