Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.4% → 87.2%

Time: 10.9s

Alternatives: 8

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: 73.4% 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: 87.2% accurate, 0.3× speedup

Math

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

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 -5e-310)
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (* (/ c0 (sqrt l)) (/ (sqrt A) (sqrt V)))))

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. times-frac 78.4%

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

   3. Applied egg-rr 78.4%

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

      1. clear-num 78.4%

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

      2. un-div-inv 78.8%

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

   5. Applied egg-rr 78.8%

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

      1. div-inv 78.4%

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

      2. clear-num 78.4%

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

      3. frac-times 76.3%

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

      4. *-un-lft-identity 76.3%

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

      5. associate-/l/ 78.5%

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

      6. frac-2neg 78.5%

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

      7. sqrt-div 88.5%

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

   7. Applied egg-rr 88.5%

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

      1. distribute-neg-frac 88.5%

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

   9. Simplified 88.5%

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

   if -4.999999999999985e-310 < V

   1. Initial program 67.4%

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

      1. *-un-lft-identity 67.4%

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

      2. times-frac 65.2%

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

   3. Applied egg-rr 65.2%

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

      1. clear-num 65.0%

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

      2. un-div-inv 65.0%

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

   5. Applied egg-rr 65.0%

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

      1. associate-/l/ 65.0%

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

      2. sqrt-div 65.4%

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

      3. metadata-eval 65.4%

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

      4. /-rgt-identity 65.4%

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

      5. clear-num 65.4%

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

      6. div-inv 65.4%

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

      7. clear-num 64.9%

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

      8. div-inv 64.9%

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

      9. clear-num 65.1%

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

      10. frac-times 67.3%

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

      11. *-un-lft-identity 67.3%

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

      12. div-inv 66.3%

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

      13. metadata-eval 66.3%

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

      14. frac-times 66.3%

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

      15. div-inv 66.3%

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

      16. associate-/l/ 66.3%

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

      17. un-div-inv 67.3%

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

      18. clear-num 67.2%

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

      19. associate-*r/ 66.2%

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

      20. clear-num 66.0%

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

      21. div-inv 66.0%

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

      22. sqrt-undiv 43.1%

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

   7. Applied egg-rr 51.4%

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

      1. times-frac 53.0%

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

   9. Simplified 53.0%

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

4. Final simplification 71.0%

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

Alternative 2: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 10^{-225}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;t_0 \leq 10^{+259}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \end{array} \]

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-225)
     (* c0 (sqrt (/ (/ A V) l)))
     (if (<= t_0 1e+259) 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-225) {
		tmp = c0 * sqrt(((A / V) / l));
	} else if (t_0 <= 1e+259) {
		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-225) then
        tmp = c0 * sqrt(((a / v) / l))
    else if (t_0 <= 1d+259) 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-225) {
		tmp = c0 * Math.sqrt(((A / V) / l));
	} else if (t_0 <= 1e+259) {
		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-225:
		tmp = c0 * math.sqrt(((A / V) / l))
	elif t_0 <= 1e+259:
		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-225)
		tmp = Float64(c0 * sqrt(Float64(Float64(A / V) / l)));
	elseif (t_0 <= 1e+259)
		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-225)
		tmp = c0 * sqrt(((A / V) / l));
	elseif (t_0 <= 1e+259)
		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-225], N[(c0 * N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+259], 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)))) < 9.9999999999999996e-226

   1. Initial program 67.3%

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

      1. associate-/r* 69.1%

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

   3. Simplified 69.1%

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

   if 9.9999999999999996e-226 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 9.999999999999999e258

   1. Initial program 99.6%

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

   if 9.999999999999999e258 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 55.0%

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

      1. associate-/r* 65.1%

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

      2. clear-num 65.1%

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

      3. sqrt-div 70.6%

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

      4. metadata-eval 70.6%

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

      5. div-inv 70.6%

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

      6. clear-num 70.6%

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

   3. Applied egg-rr 70.6%

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

      1. associate-*r/ 58.5%

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

      2. *-commutative 58.5%

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

      3. *-lft-identity 58.5%

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

      4. times-frac 68.1%

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

      5. remove-double-div 68.1%

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

      6. associate-/r* 68.1%

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

      7. *-rgt-identity 68.1%

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

      8. remove-double-div 68.1%

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

   5. Simplified 68.1%

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

   7. Applied egg-rr 57.7%

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

      1. expm1-def 57.7%

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

      2. expm1-log1p 58.5%

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

      3. associate-*r/ 68.1%

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

   9. Simplified 68.1%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{-225}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{+259}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \end{array} \]

Alternative 3: 89.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{V}}\\ \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{t_0}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-221}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+287}\right):\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

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 (sqrt (/ A V))))
   (if (<= (* V l) (- INFINITY))
     (/ t_0 (/ (sqrt l) c0))
     (if (<= (* V l) -5e-221)
       (* c0 (/ (sqrt (- A)) (sqrt (* V (- l)))))
       (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+287)))
         (/ c0 (/ (sqrt l) t_0))
         (* c0 (/ (sqrt A) (sqrt (* V l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / V));
	double tmp;
	if ((V * l) <= -((double) INFINITY)) {
		tmp = t_0 / (sqrt(l) / c0);
	} else if ((V * l) <= -5e-221) {
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+287)) {
		tmp = c0 / (sqrt(l) / t_0);
	} else {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	}
	return tmp;
}

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = Math.sqrt((A / V));
	double tmp;
	if ((V * l) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 / (Math.sqrt(l) / c0);
	} else if ((V * l) <= -5e-221) {
		tmp = c0 * (Math.sqrt(-A) / Math.sqrt((V * -l)));
	} else if (((V * l) <= 0.0) || !((V * l) <= 1e+287)) {
		tmp = c0 / (Math.sqrt(l) / t_0);
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / V))
	tmp = 0
	if (V * l) <= -math.inf:
		tmp = t_0 / (math.sqrt(l) / c0)
	elif (V * l) <= -5e-221:
		tmp = c0 * (math.sqrt(-A) / math.sqrt((V * -l)))
	elif ((V * l) <= 0.0) or not ((V * l) <= 1e+287):
		tmp = c0 / (math.sqrt(l) / t_0)
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = sqrt(Float64(A / V))
	tmp = 0.0
	if (Float64(V * l) <= Float64(-Inf))
		tmp = Float64(t_0 / Float64(sqrt(l) / c0));
	elseif (Float64(V * l) <= -5e-221)
		tmp = Float64(c0 * Float64(sqrt(Float64(-A)) / sqrt(Float64(V * Float64(-l)))));
	elseif ((Float64(V * l) <= 0.0) || !(Float64(V * l) <= 1e+287))
		tmp = Float64(c0 / Float64(sqrt(l) / t_0));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(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)
	t_0 = sqrt((A / V));
	tmp = 0.0;
	if ((V * l) <= -Inf)
		tmp = t_0 / (sqrt(l) / c0);
	elseif ((V * l) <= -5e-221)
		tmp = c0 * (sqrt(-A) / sqrt((V * -l)));
	elseif (((V * l) <= 0.0) || ~(((V * l) <= 1e+287)))
		tmp = c0 / (sqrt(l) / t_0);
	else
		tmp = c0 * (sqrt(A) / sqrt((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_] := Block[{t$95$0 = N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], (-Infinity)], N[(t$95$0 / N[(N[Sqrt[l], $MachinePrecision] / c0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], -5e-221], N[(c0 * N[(N[Sqrt[(-A)], $MachinePrecision] / N[Sqrt[N[(V * (-l)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+287]], $MachinePrecision]], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 20.6%

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

      1. associate-/r* 52.7%

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

      2. sqrt-div 50.7%

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

      3. associate-*r/ 50.6%

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

   3. Applied egg-rr 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-/l* 50.8%

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

   5. Simplified 50.8%

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

   if -inf.0 < (*.f64 V l) < -4.99999999999999996e-221

   1. Initial program 87.8%

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

      1. frac-2neg 87.8%

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

      2. sqrt-div 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. distribute-rgt-neg-out 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

   5. Simplified 99.5%

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

   if -4.99999999999999996e-221 < (*.f64 V l) < -0.0 or 1.0000000000000001e287 < (*.f64 V l)

   1. Initial program 43.9%

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

      1. associate-/r* 58.7%

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

      2. sqrt-div 47.9%

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

      3. associate-*r/ 47.9%

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

   3. Applied egg-rr 47.9%

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

      1. associate-/l* 48.0%

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

   5. Simplified 48.0%

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

   if -0.0 < (*.f64 V l) < 1.0000000000000001e287

   1. Initial program 87.8%

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

      1. sqrt-div 98.9%

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

      2. associate-*r/ 90.6%

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

   3. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-/l* 97.1%

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

      3. associate-/r/ 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\frac{\sqrt{\ell}}{c0}}\\ \mathbf{elif}\;V \cdot \ell \leq -5 \cdot 10^{-221}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{-A}}{\sqrt{V \cdot \left(-\ell\right)}}\\ \mathbf{elif}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+287}\right):\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 4: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-219}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-318}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V}} \cdot \sqrt{\frac{A}{\ell}}\\ \end{array} \end{array} \]

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 (/ c0 (sqrt (* V (/ l A))))))
   (if (<= (* V l) -5e+83)
     t_0
     (if (<= (* V l) -1e-219)
       (* c0 (sqrt (/ A (* V l))))
       (if (<= (* V l) 2e-318)
         t_0
         (if (<= (* V l) 5e+299)
           (* c0 (/ (sqrt A) (sqrt (* V l))))
           (* (/ c0 (sqrt V)) (sqrt (/ A l)))))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = c0 / sqrt((V * (l / A)));
	double tmp;
	if ((V * l) <= -5e+83) {
		tmp = t_0;
	} else if ((V * l) <= -1e-219) {
		tmp = c0 * sqrt((A / (V * l)));
	} else if ((V * l) <= 2e-318) {
		tmp = t_0;
	} else if ((V * l) <= 5e+299) {
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	} else {
		tmp = (c0 / sqrt(V)) * sqrt((A / 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) :: tmp
    t_0 = c0 / sqrt((v * (l / a)))
    if ((v * l) <= (-5d+83)) then
        tmp = t_0
    else if ((v * l) <= (-1d-219)) then
        tmp = c0 * sqrt((a / (v * l)))
    else if ((v * l) <= 2d-318) then
        tmp = t_0
    else if ((v * l) <= 5d+299) then
        tmp = c0 * (sqrt(a) / sqrt((v * l)))
    else
        tmp = (c0 / sqrt(v)) * sqrt((a / 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 = c0 / Math.sqrt((V * (l / A)));
	double tmp;
	if ((V * l) <= -5e+83) {
		tmp = t_0;
	} else if ((V * l) <= -1e-219) {
		tmp = c0 * Math.sqrt((A / (V * l)));
	} else if ((V * l) <= 2e-318) {
		tmp = t_0;
	} else if ((V * l) <= 5e+299) {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((V * l)));
	} else {
		tmp = (c0 / Math.sqrt(V)) * Math.sqrt((A / l));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = c0 / math.sqrt((V * (l / A)))
	tmp = 0
	if (V * l) <= -5e+83:
		tmp = t_0
	elif (V * l) <= -1e-219:
		tmp = c0 * math.sqrt((A / (V * l)))
	elif (V * l) <= 2e-318:
		tmp = t_0
	elif (V * l) <= 5e+299:
		tmp = c0 * (math.sqrt(A) / math.sqrt((V * l)))
	else:
		tmp = (c0 / math.sqrt(V)) * math.sqrt((A / l))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(c0 / sqrt(Float64(V * Float64(l / A))))
	tmp = 0.0
	if (Float64(V * l) <= -5e+83)
		tmp = t_0;
	elseif (Float64(V * l) <= -1e-219)
		tmp = Float64(c0 * sqrt(Float64(A / Float64(V * l))));
	elseif (Float64(V * l) <= 2e-318)
		tmp = t_0;
	elseif (Float64(V * l) <= 5e+299)
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(V * l))));
	else
		tmp = Float64(Float64(c0 / sqrt(V)) * sqrt(Float64(A / 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 = c0 / sqrt((V * (l / A)));
	tmp = 0.0;
	if ((V * l) <= -5e+83)
		tmp = t_0;
	elseif ((V * l) <= -1e-219)
		tmp = c0 * sqrt((A / (V * l)));
	elseif ((V * l) <= 2e-318)
		tmp = t_0;
	elseif ((V * l) <= 5e+299)
		tmp = c0 * (sqrt(A) / sqrt((V * l)));
	else
		tmp = (c0 / sqrt(V)) * sqrt((A / 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[(c0 / N[Sqrt[N[(V * N[(l / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(V * l), $MachinePrecision], -5e+83], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], -1e-219], N[(c0 * N[Sqrt[N[(A / N[(V * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(V * l), $MachinePrecision], 2e-318], t$95$0, If[LessEqual[N[(V * l), $MachinePrecision], 5e+299], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(V * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c0 / N[Sqrt[V], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(A / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 V l) < -5.00000000000000029e83 or -1e-219 < (*.f64 V l) < 2.0000024e-318

   1. Initial program 53.4%

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

      1. associate-/r* 65.2%

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

      2. clear-num 65.2%

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

      3. sqrt-div 65.8%

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

      4. metadata-eval 65.8%

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

      5. div-inv 65.5%

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

      6. clear-num 65.5%

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

   3. Applied egg-rr 65.5%

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

      1. associate-*r/ 53.4%

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

      2. *-commutative 53.4%

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

      3. *-lft-identity 53.4%

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

      4. times-frac 65.4%

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

      5. remove-double-div 65.4%

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

      6. associate-/r* 65.4%

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

      7. *-rgt-identity 65.4%

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

      8. remove-double-div 65.4%

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

   5. Simplified 65.4%

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

   7. Applied egg-rr 27.4%

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

      1. expm1-def 39.7%

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

      2. expm1-log1p 53.5%

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

      3. associate-*r/ 65.6%

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

   9. Simplified 65.6%

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

   if -5.00000000000000029e83 < (*.f64 V l) < -1e-219

   1. Initial program 93.3%

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

   if 2.0000024e-318 < (*.f64 V l) < 5.0000000000000003e299

   1. Initial program 87.4%

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

      1. sqrt-div 99.4%

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

      2. associate-*r/ 91.2%

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

   3. Applied egg-rr 91.2%

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

      1. *-commutative 91.2%

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

      2. associate-/l* 98.0%

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

      3. associate-/r/ 99.4%

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

   5. Simplified 99.4%

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

   if 5.0000000000000003e299 < (*.f64 V l)

   1. Initial program 28.9%

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

      1. associate-/r* 52.3%

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

      2. sqrt-div 55.7%

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

      3. associate-*r/ 55.7%

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

   3. Applied egg-rr 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-/l* 55.5%

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

   5. Simplified 55.5%

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

      1. associate-/r/ 55.7%

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

      2. sqrt-undiv 52.3%

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

      3. un-div-inv 52.3%

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

      4. frac-times 28.9%

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

      5. *-commutative 28.9%

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

      6. frac-times 52.1%

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

      7. *-commutative 52.1%

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

      8. clear-num 52.1%

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

      9. frac-times 52.3%

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

      10. add-sqr-sqrt 52.2%

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

      11. frac-times 52.2%

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

      12. sqrt-unprod 52.2%

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

      13. add-sqr-sqrt 52.2%

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

      14. *-commutative 52.2%

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

      15. associate-*r/ 52.4%

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

      16. sqrt-prod 55.7%

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

      17. times-frac 55.7%

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

      18. metadata-eval 55.7%

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

      19. sqrt-div 55.6%

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

      20. clear-num 55.6%

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

   7. Applied egg-rr 55.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq -1 \cdot 10^{-219}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \leq 2 \cdot 10^{-318}:\\ \;\;\;\;\frac{c0}{\sqrt{V \cdot \frac{\ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \leq 5 \cdot 10^{+299}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V}} \cdot \sqrt{\frac{A}{\ell}}\\ \end{array} \]

Alternative 5: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+287}\right):\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (if (or (<= (* V l) 0.0) (not (<= (* V l) 1e+287)))
   (/ c0 (/ (sqrt l) (sqrt (/ A V))))
   (* c0 (/ (sqrt A) (sqrt (* 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) <= 0.0) || !((V * l) <= 1e+287)) {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	} else {
		tmp = c0 * (sqrt(A) / sqrt((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) <= 0.0d0) .or. (.not. ((v * l) <= 1d+287))) then
        tmp = c0 / (sqrt(l) / sqrt((a / v)))
    else
        tmp = c0 * (sqrt(a) / sqrt((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) <= 0.0) || !((V * l) <= 1e+287)) {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	} else {
		tmp = c0 * (Math.sqrt(A) / Math.sqrt((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) <= 0.0) or not ((V * l) <= 1e+287):
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	else:
		tmp = c0 * (math.sqrt(A) / math.sqrt((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) <= 0.0) || !(Float64(V * l) <= 1e+287))
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	else
		tmp = Float64(c0 * Float64(sqrt(A) / sqrt(Float64(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) <= 0.0) || ~(((V * l) <= 1e+287)))
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	else
		tmp = c0 * (sqrt(A) / sqrt((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[Or[LessEqual[N[(V * l), $MachinePrecision], 0.0], N[Not[LessEqual[N[(V * l), $MachinePrecision], 1e+287]], $MachinePrecision]], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $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 (*.f64 V l) < -0.0 or 1.0000000000000001e287 < (*.f64 V l)

   1. Initial program 61.8%

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

      1. associate-/r* 67.5%

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

      2. sqrt-div 45.8%

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

      3. associate-*r/ 45.6%

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

   3. Applied egg-rr 45.6%

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

      1. associate-/l* 45.8%

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

   5. Simplified 45.8%

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

   if -0.0 < (*.f64 V l) < 1.0000000000000001e287

   1. Initial program 87.8%

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

      1. sqrt-div 98.9%

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

      2. associate-*r/ 90.6%

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

   3. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-/l* 97.1%

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

      3. associate-/r/ 98.9%

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

   5. Simplified 98.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \leq 0 \lor \neg \left(V \cdot \ell \leq 10^{+287}\right):\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array} \]

Alternative 6: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{if}\;t_0 \leq 10^{-225}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

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 (<= t_0 1e-225) (* 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 <= 1e-225) {
		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 <= 1d-225) 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 <= 1e-225) {
		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 <= 1e-225:
		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 <= 1e-225)
		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 <= 1e-225)
		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[LessEqual[t$95$0, 1e-225], 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)))) < 9.9999999999999996e-226

   1. Initial program 67.3%

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

      1. associate-/r* 69.1%

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

   3. Simplified 69.1%

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

   if 9.9999999999999996e-226 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 81.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \leq 10^{-225}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \end{array} \]

Alternative 7: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \leq 4 \cdot 10^{+22}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \end{array} \end{array} \]

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 -5e-310)
   (* c0 (/ (sqrt (/ (- A) l)) (sqrt (- V))))
   (if (<= V 4e+22)
     (* c0 (* (sqrt A) (sqrt (/ (/ 1.0 V) l))))
     (/ c0 (/ (sqrt l) (sqrt (/ A V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double tmp;
	if (V <= -5e-310) {
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	} else if (V <= 4e+22) {
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	}
	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 <= (-5d-310)) then
        tmp = c0 * (sqrt((-a / l)) / sqrt(-v))
    else if (v <= 4d+22) then
        tmp = c0 * (sqrt(a) * sqrt(((1.0d0 / v) / l)))
    else
        tmp = c0 / (sqrt(l) / sqrt((a / v)))
    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 <= -5e-310) {
		tmp = c0 * (Math.sqrt((-A / l)) / Math.sqrt(-V));
	} else if (V <= 4e+22) {
		tmp = c0 * (Math.sqrt(A) * Math.sqrt(((1.0 / V) / l)));
	} else {
		tmp = c0 / (Math.sqrt(l) / Math.sqrt((A / V)));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	tmp = 0
	if V <= -5e-310:
		tmp = c0 * (math.sqrt((-A / l)) / math.sqrt(-V))
	elif V <= 4e+22:
		tmp = c0 * (math.sqrt(A) * math.sqrt(((1.0 / V) / l)))
	else:
		tmp = c0 / (math.sqrt(l) / math.sqrt((A / V)))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	tmp = 0.0
	if (V <= -5e-310)
		tmp = Float64(c0 * Float64(sqrt(Float64(Float64(-A) / l)) / sqrt(Float64(-V))));
	elseif (V <= 4e+22)
		tmp = Float64(c0 * Float64(sqrt(A) * sqrt(Float64(Float64(1.0 / V) / l))));
	else
		tmp = Float64(c0 / Float64(sqrt(l) / sqrt(Float64(A / V))));
	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 <= -5e-310)
		tmp = c0 * (sqrt((-A / l)) / sqrt(-V));
	elseif (V <= 4e+22)
		tmp = c0 * (sqrt(A) * sqrt(((1.0 / V) / l)));
	else
		tmp = c0 / (sqrt(l) / sqrt((A / V)));
	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[V, -5e-310], N[(c0 * N[(N[Sqrt[N[((-A) / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[V, 4e+22], N[(c0 * N[(N[Sqrt[A], $MachinePrecision] * N[Sqrt[N[(N[(1.0 / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c0 / N[(N[Sqrt[l], $MachinePrecision] / N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if V < -4.999999999999985e-310

   1. Initial program 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. times-frac 78.4%

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

   3. Applied egg-rr 78.4%

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

      1. clear-num 78.4%

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

      2. un-div-inv 78.8%

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

   5. Applied egg-rr 78.8%

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

      1. div-inv 78.4%

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

      2. clear-num 78.4%

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

      3. frac-times 76.3%

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

      4. *-un-lft-identity 76.3%

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

      5. associate-/l/ 78.5%

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

      6. frac-2neg 78.5%

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

      7. sqrt-div 88.5%

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

   7. Applied egg-rr 88.5%

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

      1. distribute-neg-frac 88.5%

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

   9. Simplified 88.5%

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

   if -4.999999999999985e-310 < V < 4e22

   1. Initial program 76.0%

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

      1. div-inv 74.1%

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

      2. sqrt-prod 49.4%

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

      3. associate-/r* 49.3%

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

   3. Applied egg-rr 49.3%

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

   if 4e22 < V

   1. Initial program 56.3%

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

      1. associate-/r* 59.9%

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

      2. sqrt-div 42.2%

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

      3. associate-*r/ 40.6%

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

   3. Applied egg-rr 40.6%

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

      1. associate-/l* 42.3%

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

   5. Simplified 42.3%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \leq -5 \cdot 10^{-310}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{\frac{-A}{\ell}}}{\sqrt{-V}}\\ \mathbf{elif}\;V \leq 4 \cdot 10^{+22}:\\ \;\;\;\;c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{\frac{1}{V}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\frac{\sqrt{\ell}}{\sqrt{\frac{A}{V}}}}\\ \end{array} \]

Alternative 8: 73.4% 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 71.9%

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

2. Final simplification 71.9%

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

Reproduce

herbie shell --seed 2023318 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))