Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 90.9%

Time: 9.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 0.3× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* c0 (/ (/ (sqrt (- A)) (sqrt (- V))) (sqrt l)))
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((sqrt(-A) / sqrt(-V)) / sqrt(l));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-5d-310)) then
        tmp = c0 * ((sqrt(-a) / sqrt(-v)) / sqrt(l))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * ((Math.sqrt(-A) / Math.sqrt(-V)) / Math.sqrt(l));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * ((math.sqrt(-A) / math.sqrt(-V)) / math.sqrt(l))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(Float64(sqrt(Float64(-A)) / sqrt(Float64(-V))) / sqrt(l)));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 * N[(N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 78.0%

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

      1. associate-/r* 76.8%

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

      2. sqrt-div 46.2%

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

   3. Applied egg-rr 46.2%

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

      1. frac-2neg 46.2%

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

      2. sqrt-div 53.0%

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

   5. Applied egg-rr 53.0%

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

   if -4.999999999999985e-310 < A

   1. Initial program 77.8%

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

      1. sqrt-div 87.0%

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

      2. associate-*r/ 82.5%

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

   3. Applied egg-rr 82.5%

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

      1. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

4. Final simplification 69.4%

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

Alternative 2: 84.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-5d-310)) then
        tmp = c0 * (sqrt(-a) / sqrt((v * -l)))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 78.0%

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

      1. frac-2neg 78.0%

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

      2. sqrt-div 86.1%

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

      3. *-commutative 86.1%

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

      4. distribute-rgt-neg-in 86.1%

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

   3. Applied egg-rr 86.1%

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

   if -4.999999999999985e-310 < A

   1. Initial program 77.8%

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

      1. sqrt-div 87.0%

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

      2. associate-*r/ 82.5%

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

   3. Applied egg-rr 82.5%

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

      1. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

4. Final simplification 86.6%

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

Alternative 3: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l 1.4e-143)
   (/ c0 (sqrt (* V (/ l A))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 1.4e-143) {
		tmp = c0 / sqrt((V * (l / A)));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: 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 (l <= 1.4d-143) then
        tmp = c0 / sqrt((v * (l / a)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= 1.4e-143) {
		tmp = c0 / Math.sqrt((V * (l / A)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= 1.4e-143:
		tmp = c0 / math.sqrt((V * (l / A)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= 1.4e-143)
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	tmp = 0.0;
	if (l <= 1.4e-143)
		tmp = c0 / sqrt((V * (l / A)));
	else
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, 1.4e-143], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3999999999999999e-143

   1. Initial program 75.9%

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

      1. *-un-lft-identity 75.9%

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

      2. times-frac 76.0%

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

   3. Applied egg-rr 76.0%

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

      1. frac-times 75.9%

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

      2. *-un-lft-identity 75.9%

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

      3. sqrt-div 38.8%

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

      4. clear-num 38.8%

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

      5. div-inv 38.8%

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

      6. sqrt-undiv 76.1%

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

      7. associate-*r/ 76.6%

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

   5. Applied egg-rr 76.6%

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

   if 1.3999999999999999e-143 < l

   1. Initial program 80.9%

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

      1. associate-/r* 82.8%

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

      2. sqrt-div 91.3%

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

   3. Applied egg-rr 91.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{-143}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array} \]

Alternative 4: 83.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= l -5e-310)
   (* (sqrt A) (/ c0 (sqrt (* V l))))
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = sqrt(A) * (c0 / sqrt((V * l)));
	} else {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	}
	return tmp;
}

Fortran

NOTE: 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 (l <= (-5d-310)) then
        tmp = sqrt(a) * (c0 / sqrt((v * l)))
    else
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (l <= -5e-310) {
		tmp = Math.sqrt(A) * (c0 / Math.sqrt((V * l)));
	} else {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if l <= -5e-310:
		tmp = math.sqrt(A) * (c0 / math.sqrt((V * l)))
	else:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(sqrt(A) * Float64(c0 / sqrt(Float64(V * l))));
	else
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[l, -5e-310], N[(N[Sqrt[A], $MachinePrecision] * N[(c0 / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 75.5%

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

      1. sqrt-div 36.9%

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

      2. associate-*r/ 36.9%

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

   3. Applied egg-rr 36.9%

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

      1. associate-*l/ 36.1%

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

   5. Simplified 36.1%

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

   if -4.999999999999985e-310 < l

   1. Initial program 79.9%

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

      1. associate-/r* 79.9%

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

      2. sqrt-div 88.8%

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

   3. Applied egg-rr 88.8%

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

4. Final simplification 64.7%

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

Alternative 5: 84.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (* c0 (/ (sqrt (/ A V)) (sqrt l)))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (sqrt((A / V)) / sqrt(l));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-5d-310)) then
        tmp = c0 * (sqrt((a / v)) / sqrt(l))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 * (Math.sqrt((A / V)) / Math.sqrt(l));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 * (math.sqrt((A / V)) / math.sqrt(l))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(A / V)) / sqrt(l)));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 * N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 78.0%

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

      1. associate-/r* 76.8%

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

      2. sqrt-div 46.2%

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

   3. Applied egg-rr 46.2%

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

   if -4.999999999999985e-310 < A

   1. Initial program 77.8%

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

      1. sqrt-div 87.0%

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

      2. associate-*r/ 82.5%

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

   3. Applied egg-rr 82.5%

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

      1. *-commutative 82.5%

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

      2. associate-*l/ 87.0%

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

   5. Simplified 87.0%

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

4. Final simplification 65.8%

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

Alternative 6: 84.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (* c0 (/ (sqrt A) (sqrt (* V l))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-5d-310)) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 78.0%

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

      1. *-un-lft-identity 78.0%

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

      2. times-frac 75.4%

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

   3. Applied egg-rr 75.4%

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

      1. frac-times 78.0%

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

      2. *-un-lft-identity 78.0%

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

      3. sqrt-div 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. div-inv 0.0%

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

      6. sqrt-undiv 77.7%

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

      7. associate-*r/ 73.7%

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

   5. Applied egg-rr 73.7%

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

   6. Taylor expanded in V around 0 77.7%

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

      1. associate-*l/ 76.7%

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

      2. sqrt-prod 46.2%

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

   8. Applied egg-rr 46.2%

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

   if -4.999999999999985e-310 < A

   1. Initial program 77.8%

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

      1. sqrt-div 87.0%

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

      2. associate-*r/ 82.5%

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

   3. Applied egg-rr 82.5%

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

      1. *-commutative 82.5%

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

      2. associate-*l/ 87.0%

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

   5. Simplified 87.0%

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

4. Final simplification 65.8%

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

Alternative 7: 84.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: V and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (<= A -5e-310)
   (/ c0 (* (sqrt l) (sqrt (/ V A))))
   (/ c0 (/ (sqrt (* V l)) (sqrt A)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 / (sqrt(l) * sqrt((V / A)));
	} else {
		tmp = c0 / (sqrt((V * l)) / sqrt(A));
	}
	return tmp;
}

Fortran

NOTE: 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 <= (-5d-310)) then
        tmp = c0 / (sqrt(l) * sqrt((v / a)))
    else
        tmp = c0 / (sqrt((v * l)) / sqrt(a))
    end if
    code = tmp
end function

Java

assert V < l;
public static double code(double c0, double A, double V, double l) {
	double tmp;
	if (A <= -5e-310) {
		tmp = c0 / (Math.sqrt(l) * Math.sqrt((V / A)));
	} else {
		tmp = c0 / (Math.sqrt((V * l)) / Math.sqrt(A));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	tmp = 0
	if A <= -5e-310:
		tmp = c0 / (math.sqrt(l) * math.sqrt((V / A)))
	else:
		tmp = c0 / (math.sqrt((V * l)) / math.sqrt(A))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (A <= -5e-310)
		tmp = Float64(c0 / Float64(sqrt(l) * sqrt(Float64(V / A))));
	else
		tmp = Float64(c0 / Float64(sqrt(Float64(V * l)) / sqrt(A)));
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: V and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := If[LessEqual[A, -5e-310], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[N[(V / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[A], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -4.999999999999985e-310

   1. Initial program 78.0%

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

      1. *-un-lft-identity 78.0%

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

      2. times-frac 75.4%

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

   3. Applied egg-rr 75.4%

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

      1. frac-times 78.0%

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

      2. *-un-lft-identity 78.0%

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

      3. sqrt-div 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. div-inv 0.0%

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

      6. sqrt-undiv 77.7%

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

      7. associate-*r/ 73.7%

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

   5. Applied egg-rr 73.7%

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

   6. Taylor expanded in V around 0 77.7%

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

      1. associate-*l/ 76.7%

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

      2. sqrt-prod 46.2%

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

   8. Applied egg-rr 46.2%

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

   if -4.999999999999985e-310 < A

   1. Initial program 77.8%

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

      1. sqrt-div 87.0%

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

      2. associate-*r/ 82.5%

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

   3. Applied egg-rr 82.5%

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

      1. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

4. Final simplification 65.8%

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

Alternative 8: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0 \lor \neg \left(t_0 \leq 5 \cdot 10^{+257}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 0.0) (not (<= t_0 5e+257)))
     (* c0 (sqrt (/ (/ A V) l)))
     (* c0 (sqrt t_0)))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= 5e+257)) {
		tmp = c0 * sqrt(((A / V) / l));
	} else {
		tmp = c0 * sqrt(t_0);
	}
	return tmp;
}

Fortran

NOTE: 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 <= 0.0d0) .or. (.not. (t_0 <= 5d+257))) then
        tmp = c0 * sqrt(((a / v) / l))
    else
        tmp = c0 * sqrt(t_0)
    end if
    code = tmp
end function

Java

assert 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 <= 0.0) || !(t_0 <= 5e+257)) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else {
		tmp = c0 * Math.sqrt(t_0);
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= 5e+257):
		tmp = c0 * math.sqrt(((A / V) / l))
	else:
		tmp = c0 * math.sqrt(t_0)
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= 5e+257))
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	else
		tmp = Float64(c0 * sqrt(t_0));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= 5e+257)))
		tmp = c0 * sqrt(((A / V) / l));
	else
		tmp = c0 * sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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, 0.0], N[Not[LessEqual[t$95$0, 5e+257]], $MachinePrecision]], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / 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)) < 0.0 or 5.00000000000000028e257 < (/.f64 A (*.f64 V l))

   1. Initial program 48.2%

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

      1. pow1/2 48.2%

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

      2. clear-num 48.2%

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

      3. inv-pow 48.2%

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

      4. pow-pow 48.6%

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

      5. associate-/l* 61.3%

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

      6. metadata-eval 61.3%

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

   3. Applied egg-rr 61.3%

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

      1. associate-/l* 48.6%

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

   5. Simplified 48.6%

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

   6. Taylor expanded in c0 around 0 48.2%

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

      1. associate-/r* 59.1%

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

   8. Simplified 59.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5.00000000000000028e257

   1. Initial program 99.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0 \lor \neg \left(\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+257}\right):\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 9: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 0.0)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+307) (* c0 (sqrt t_0)) (/ c0 (sqrt (* V (/ l A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+307) {
		tmp = c0 * sqrt(t_0);
	} else {
		tmp = c0 / sqrt((V * (l / A)));
	}
	return tmp;
}

Fortran

NOTE: 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 <= 0.0d0) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+307) then
        tmp = c0 * sqrt(t_0)
    else
        tmp = c0 / sqrt((v * (l / a)))
    end if
    code = tmp
end function

Java

assert 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 <= 0.0) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+307) {
		tmp = c0 * Math.sqrt(t_0);
	} else {
		tmp = c0 / Math.sqrt((V * (l / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 0.0:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+307:
		tmp = c0 * math.sqrt(t_0)
	else:
		tmp = c0 / math.sqrt((V * (l / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = 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 <= 5e+307)
		tmp = Float64(c0 * sqrt(t_0));
	else
		tmp = Float64(c0 / sqrt(Float64(V * Float64(l / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+307)
		tmp = c0 * sqrt(t_0);
	else
		tmp = c0 / sqrt((V * (l / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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, 0.0], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+307], N[(c0 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 47.6%

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

      1. pow1/2 47.6%

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

      2. clear-num 47.6%

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

      3. inv-pow 47.6%

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

      4. pow-pow 47.6%

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

      5. associate-/l* 64.1%

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

      6. metadata-eval 64.1%

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

   3. Applied egg-rr 64.1%

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

      1. associate-/l* 47.6%

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

   5. Simplified 47.6%

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

   6. Taylor expanded in c0 around 0 47.6%

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

      1. associate-/r* 64.1%

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

   8. Simplified 64.1%

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

   if 0.0 < (/.f64 A (*.f64 V l)) < 5e307

   1. Initial program 99.3%

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

   if 5e307 < (/.f64 A (*.f64 V l))

   1. Initial program 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. times-frac 54.0%

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

   3. Applied egg-rr 54.0%

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

      1. frac-times 40.3%

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

      2. *-un-lft-identity 40.3%

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

      3. sqrt-div 37.8%

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

      4. clear-num 37.7%

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

      5. div-inv 37.8%

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

      6. sqrt-undiv 41.1%

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

      7. associate-*r/ 54.8%

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

   5. Applied egg-rr 54.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 10: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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 2e-319)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 5e+257)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ c0 (sqrt (* l (/ V A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 2e-319) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 5e+257) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: 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 <= 2d-319) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 5d+257) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert 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 <= 2e-319) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 5e+257) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 2e-319:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 5e+257:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 2e-319)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 5e+257)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 2e-319)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 5e+257)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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, 2e-319], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+257], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 1.99998e-319

   1. Initial program 47.6%

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

      1. pow1/2 47.6%

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

      2. clear-num 47.0%

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

      3. inv-pow 47.0%

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

      4. pow-pow 47.0%

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

      5. associate-/l* 63.1%

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

      6. metadata-eval 63.1%

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

   3. Applied egg-rr 63.1%

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

      1. associate-/l* 47.0%

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

   5. Simplified 47.0%

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

   6. Taylor expanded in c0 around 0 47.6%

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

      1. associate-/r* 63.7%

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

   8. Simplified 63.7%

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

   if 1.99998e-319 < (/.f64 A (*.f64 V l)) < 5.00000000000000028e257

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. times-frac 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. frac-times 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. sqrt-div 48.4%

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

      4. clear-num 48.4%

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

      5. div-inv 48.4%

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

      6. sqrt-undiv 99.6%

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

      7. associate-*r/ 87.3%

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

   5. Applied egg-rr 87.3%

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

   6. Taylor expanded in V around 0 99.6%

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

   if 5.00000000000000028e257 < (/.f64 A (*.f64 V l))

   1. Initial program 48.5%

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

      1. *-un-lft-identity 48.5%

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

      2. times-frac 58.9%

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

   3. Applied egg-rr 58.9%

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

      1. expm1-log1p-u 31.5%

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

      2. expm1-udef 25.1%

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

      3. frac-times 23.6%

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

      4. *-un-lft-identity 23.6%

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

      5. sqrt-div 17.3%

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

      6. clear-num 17.3%

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

      7. div-inv 17.3%

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

      8. sqrt-undiv 23.6%

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

      9. associate-*r/ 25.1%

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

   5. Applied egg-rr 25.1%

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

      1. expm1-def 31.5%

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

      2. expm1-log1p 59.5%

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

      3. *-commutative 59.5%

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

      4. associate-/r/ 56.6%

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

      5. associate-/l* 49.1%

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

      6. associate-*r/ 56.6%

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

   7. Simplified 56.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 2 \cdot 10^{-319}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 11: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [V, l] = \mathsf{sort}([V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{V \cdot \ell}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \end{array} \]

FPCore

NOTE: 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-308)
     (* c0 (sqrt (* (/ 1.0 V) (/ A l))))
     (if (<= t_0 5e+257)
       (/ c0 (sqrt (/ (* V l) A)))
       (/ c0 (sqrt (* l (/ V A))))))))

C

assert(V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (V * l);
	double tmp;
	if (t_0 <= 5e-308) {
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	} else if (t_0 <= 5e+257) {
		tmp = c0 / sqrt(((V * l) / A));
	} else {
		tmp = c0 / sqrt((l * (V / A)));
	}
	return tmp;
}

Fortran

NOTE: 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-308) then
        tmp = c0 * sqrt(((1.0d0 / v) * (a / l)))
    else if (t_0 <= 5d+257) then
        tmp = c0 / sqrt(((v * l) / a))
    else
        tmp = c0 / sqrt((l * (v / a)))
    end if
    code = tmp
end function

Java

assert 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-308) {
		tmp = c0 * Math.sqrt(((1.0 / V) * (A / l)));
	} else if (t_0 <= 5e+257) {
		tmp = c0 / Math.sqrt(((V * l) / A));
	} else {
		tmp = c0 / Math.sqrt((l * (V / A)));
	}
	return tmp;
}

Python

[V, l] = sort([V, l])
def code(c0, A, V, l):
	t_0 = A / (V * l)
	tmp = 0
	if t_0 <= 5e-308:
		tmp = c0 * math.sqrt(((1.0 / V) * (A / l)))
	elif t_0 <= 5e+257:
		tmp = c0 / math.sqrt(((V * l) / A))
	else:
		tmp = c0 / math.sqrt((l * (V / A)))
	return tmp

Julia

V, l = sort([V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(V * l))
	tmp = 0.0
	if (t_0 <= 5e-308)
		tmp = Float64(c0 * sqrt(Float64(Float64(1.0 / V) * Float64(A / l))));
	elseif (t_0 <= 5e+257)
		tmp = Float64(c0 / sqrt(Float64(Float64(V * l) / A)));
	else
		tmp = Float64(c0 / sqrt(Float64(l * Float64(V / A))));
	end
	return tmp
end

MATLAB

V, l = num2cell(sort([V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (V * l);
	tmp = 0.0;
	if (t_0 <= 5e-308)
		tmp = c0 * sqrt(((1.0 / V) * (A / l)));
	elseif (t_0 <= 5e+257)
		tmp = c0 / sqrt(((V * l) / A));
	else
		tmp = c0 / sqrt((l * (V / A)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: 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-308], N[(c0 * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] * N[(A / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+257], N[(c0 / N[Sqrt[N[(N[(V * l), $MachinePrecision] / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(c0 / N[Sqrt[N[(l * N[(V / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 A (*.f64 V l)) < 4.99999999999999955e-308

   1. Initial program 48.8%

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

      1. *-un-lft-identity 48.8%

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

      2. times-frac 64.5%

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

   3. Applied egg-rr 64.5%

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

   if 4.99999999999999955e-308 < (/.f64 A (*.f64 V l)) < 5.00000000000000028e257

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. times-frac 88.1%

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

   3. Applied egg-rr 88.1%

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

      1. frac-times 99.6%

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

      2. *-un-lft-identity 99.6%

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

      3. sqrt-div 48.7%

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

      4. clear-num 48.7%

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

      5. div-inv 48.8%

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

      6. sqrt-undiv 99.6%

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

      7. associate-*r/ 87.2%

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

   5. Applied egg-rr 87.2%

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

   6. Taylor expanded in V around 0 99.6%

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

   if 5.00000000000000028e257 < (/.f64 A (*.f64 V l))

   1. Initial program 48.5%

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

      1. *-un-lft-identity 48.5%

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

      2. times-frac 58.9%

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

   3. Applied egg-rr 58.9%

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

      1. expm1-log1p-u 31.5%

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

      2. expm1-udef 25.1%

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

      3. frac-times 23.6%

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

      4. *-un-lft-identity 23.6%

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

      5. sqrt-div 17.3%

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

      6. clear-num 17.3%

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

      7. div-inv 17.3%

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

      8. sqrt-undiv 23.6%

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

      9. associate-*r/ 25.1%

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

   5. Applied egg-rr 25.1%

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

      1. expm1-def 31.5%

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

      2. expm1-log1p 59.5%

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

      3. *-commutative 59.5%

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

      4. associate-/r/ 56.6%

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

      5. associate-/l* 49.1%

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

      6. associate-*r/ 56.6%

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

   7. Simplified 56.6%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{-308}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\\ \mathbf{elif}\;\frac{A}{V \cdot \ell} \leq 5 \cdot 10^{+257}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\ell \cdot \frac{V}{A}}}\\ \end{array} \]

Alternative 12: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: 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 77.9%

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

2. Final simplification 77.9%

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

Reproduce

herbie shell --seed 2023273 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))