Henrywood and Agarwal, Equation (3)

Percentage Accurate: 73.7% → 90.6%

Time: 8.1s

Alternatives: 10

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: 90.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\sqrt{-A} \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{-V}}\right) \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (/ l A) V)))))
   (if (<= (* l V) -2e-311)
     (* (* (sqrt (- A)) (/ (sqrt (/ 1.0 l)) (sqrt (- V)))) c0)
     (if (<= (* l V) 0.0)
       t_0
       (if (<= (* l V) 1e+275) (/ (* (sqrt A) c0) (sqrt (* l V))) 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(((l / A) * V));
	double tmp;
	if ((l * V) <= -2e-311) {
		tmp = (sqrt(-A) * (sqrt((1.0 / l)) / sqrt(-V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	} 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(((l / a) * v))
    if ((l * v) <= (-2d-311)) then
        tmp = (sqrt(-a) * (sqrt((1.0d0 / l)) / sqrt(-v))) * c0
    else if ((l * v) <= 0.0d0) then
        tmp = t_0
    else if ((l * v) <= 1d+275) then
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    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(((l / A) * V));
	double tmp;
	if ((l * V) <= -2e-311) {
		tmp = (Math.sqrt(-A) * (Math.sqrt((1.0 / l)) / Math.sqrt(-V))) * c0;
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	} 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(((l / A) * V))
	tmp = 0
	if (l * V) <= -2e-311:
		tmp = (math.sqrt(-A) * (math.sqrt((1.0 / l)) / math.sqrt(-V))) * c0
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 1e+275:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	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(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(l * V) <= -2e-311)
		tmp = Float64(Float64(sqrt(Float64(-A)) * Float64(sqrt(Float64(1.0 / l)) / sqrt(Float64(-V)))) * c0);
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 1e+275)
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	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(((l / A) * V));
	tmp = 0.0;
	if ((l * V) <= -2e-311)
		tmp = (sqrt(-A) * (sqrt((1.0 / l)) / sqrt(-V))) * c0;
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 1e+275)
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	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[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -2e-311], N[(N[(N[Sqrt[(-A)], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / l), $MachinePrecision]], $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 1e+275], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -1.9999999999999e-311

   1. Initial program 77.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.2

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

   4. Applied rewrites 77.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lift-/.f64 N/A

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

      9. sqrt-unprod N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lift-sqrt.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-2neg N/A

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

      16. sqrt-div N/A

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

      17. associate-/r/ N/A

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

      18. lower-*.f64 N/A

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

   6. Applied rewrites 48.2%

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

   7. Taylor expanded in l around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 48.2

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

   9. Applied rewrites 48.2%

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

   if -1.9999999999999e-311 < (*.f64 V l) < 0.0 or 9.9999999999999996e274 < (*.f64 V l)

   1. Initial program 39.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 64.1

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

   4. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. lift-sqrt.f64 N/A

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

      17. div-inv N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lift-sqrt.f64 N/A

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

   6. Applied rewrites 65.3%

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

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

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 99.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\left(\sqrt{-A} \cdot \frac{\sqrt{\frac{1}{\ell}}}{\sqrt{-V}}\right) \cdot c0\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+218}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ A (* l V))) c0)))
   (if (<= t_0 0.0)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= t_0 2e+218) t_0 (/ c0 (sqrt (* (/ l A) V)))))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 2e+218) {
		tmp = t_0;
	} else {
		tmp = c0 / sqrt(((l / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((a / (l * v))) * c0
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((a / l) / v)) * c0
    else if (t_0 <= 2d+218) then
        tmp = t_0
    else
        tmp = c0 / sqrt(((l / 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 t_0 = Math.sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 2e+218) {
		tmp = t_0;
	} else {
		tmp = c0 / Math.sqrt(((l / A) * V));
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (l * V))) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif t_0 <= 2e+218:
		tmp = t_0
	else:
		tmp = c0 / math.sqrt(((l / A) * V))
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(l * V))) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (t_0 <= 2e+218)
		tmp = t_0;
	else
		tmp = Float64(c0 / sqrt(Float64(Float64(l / 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)
	t_0 = sqrt((A / (l * V))) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif (t_0 <= 2e+218)
		tmp = t_0;
	else
		tmp = c0 / sqrt(((l / 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_] := Block[{t$95$0 = N[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 2e+218], t$95$0, N[(c0 / N[Sqrt[N[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.5

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

   4. Applied rewrites 69.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 2.00000000000000017e218

   1. Initial program 98.9%

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

   if 2.00000000000000017e218 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 68.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. lift-sqrt.f64 N/A

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

      17. div-inv N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lift-sqrt.f64 N/A

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

   6. Applied rewrites 75.9%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 2 \cdot 10^{+218}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;t\_0 \leq 10^{+289}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \end{array} \]

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 (* l V))) c0)))
   (if (<= t_0 0.0)
     (* (sqrt (/ (/ A l) V)) c0)
     (if (<= t_0 1e+289) t_0 (* (sqrt (/ (/ A V) l)) c0)))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 1e+289) {
		tmp = t_0;
	} else {
		tmp = sqrt(((A / V) / l)) * c0;
	}
	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 = sqrt((a / (l * v))) * c0
    if (t_0 <= 0.0d0) then
        tmp = sqrt(((a / l) / v)) * c0
    else if (t_0 <= 1d+289) then
        tmp = t_0
    else
        tmp = sqrt(((a / v) / l)) * c0
    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 = Math.sqrt((A / (l * V))) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.sqrt(((A / l) / V)) * c0;
	} else if (t_0 <= 1e+289) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(((A / V) / l)) * c0;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = math.sqrt((A / (l * V))) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.sqrt(((A / l) / V)) * c0
	elif t_0 <= 1e+289:
		tmp = t_0
	else:
		tmp = math.sqrt(((A / V) / l)) * c0
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(sqrt(Float64(A / Float64(l * V))) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(A / l) / V)) * c0);
	elseif (t_0 <= 1e+289)
		tmp = t_0;
	else
		tmp = Float64(sqrt(Float64(Float64(A / V) / l)) * c0);
	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 / (l * V))) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = sqrt(((A / l) / V)) * c0;
	elseif (t_0 <= 1e+289)
		tmp = t_0;
	else
		tmp = sqrt(((A / V) / l)) * c0;
	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[(N[Sqrt[N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[Sqrt[N[(N[(A / l), $MachinePrecision] / V), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision], If[LessEqual[t$95$0, 1e+289], t$95$0, N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 65.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.5

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

   4. Applied rewrites 69.5%

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

   if 0.0 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l)))) < 1.0000000000000001e289

   1. Initial program 99.0%

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

   if 1.0000000000000001e289 < (*.f64 c0 (sqrt.f64 (/.f64 A (*.f64 V l))))

   1. Initial program 64.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 71.3

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

   4. Applied rewrites 71.3%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\frac{A}{\ell}}{V}} \cdot c0\\ \mathbf{elif}\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0 \leq 10^{+289}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{A}{\ell \cdot V}\\ t_1 := \sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\sqrt{t\_0} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
(FPCore (c0 A V l)
 :precision binary64
 (let* ((t_0 (/ A (* l V))) (t_1 (* (sqrt (/ (/ A V) l)) c0)))
   (if (<= t_0 0.0) t_1 (if (<= t_0 2e+276) (* (sqrt t_0) c0) t_1))))

C

assert(c0 < A && A < V && V < l);
double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+276) {
		tmp = sqrt(t_0) * c0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
real(8) function code(c0, a, v, l)
    real(8), intent (in) :: c0
    real(8), intent (in) :: a
    real(8), intent (in) :: v
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (l * v)
    t_1 = sqrt(((a / v) / l)) * c0
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 2d+276) then
        tmp = sqrt(t_0) * c0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert c0 < A && A < V && V < l;
public static double code(double c0, double A, double V, double l) {
	double t_0 = A / (l * V);
	double t_1 = Math.sqrt(((A / V) / l)) * c0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+276) {
		tmp = Math.sqrt(t_0) * c0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[c0, A, V, l] = sort([c0, A, V, l])
def code(c0, A, V, l):
	t_0 = A / (l * V)
	t_1 = math.sqrt(((A / V) / l)) * c0
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 2e+276:
		tmp = math.sqrt(t_0) * c0
	else:
		tmp = t_1
	return tmp

Julia

c0, A, V, l = sort([c0, A, V, l])
function code(c0, A, V, l)
	t_0 = Float64(A / Float64(l * V))
	t_1 = Float64(sqrt(Float64(Float64(A / V) / l)) * c0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+276)
		tmp = Float64(sqrt(t_0) * c0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

c0, A, V, l = num2cell(sort([c0, A, V, l])){:}
function tmp_2 = code(c0, A, V, l)
	t_0 = A / (l * V);
	t_1 = sqrt(((A / V) / l)) * c0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 2e+276)
		tmp = sqrt(t_0) * c0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: c0, A, V, and l should be sorted in increasing order before calling this function.
code[c0_, A_, V_, l_] := Block[{t$95$0 = N[(A / N[(l * V), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(A / V), $MachinePrecision] / l), $MachinePrecision]], $MachinePrecision] * c0), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 2e+276], N[(N[Sqrt[t$95$0], $MachinePrecision] * c0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 43.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 60.0

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

   4. Applied rewrites 60.0%

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

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

   1. Initial program 98.3%

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

4. Final simplification 81.1%

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

Alternative 5: 87.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{\sqrt{-A} \cdot c0}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (/ l A) V)))))
   (if (<= (* l V) -5e-278)
     (/ (/ (* (sqrt (- A)) c0) (sqrt (- V))) (sqrt l))
     (if (<= (* l V) 0.0)
       t_0
       (if (<= (* l V) 1e+275) (/ (* (sqrt A) c0) (sqrt (* l V))) 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(((l / A) * V));
	double tmp;
	if ((l * V) <= -5e-278) {
		tmp = ((sqrt(-A) * c0) / sqrt(-V)) / sqrt(l);
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	} 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(((l / a) * v))
    if ((l * v) <= (-5d-278)) then
        tmp = ((sqrt(-a) * c0) / sqrt(-v)) / sqrt(l)
    else if ((l * v) <= 0.0d0) then
        tmp = t_0
    else if ((l * v) <= 1d+275) then
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    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(((l / A) * V));
	double tmp;
	if ((l * V) <= -5e-278) {
		tmp = ((Math.sqrt(-A) * c0) / Math.sqrt(-V)) / Math.sqrt(l);
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	} 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(((l / A) * V))
	tmp = 0
	if (l * V) <= -5e-278:
		tmp = ((math.sqrt(-A) * c0) / math.sqrt(-V)) / math.sqrt(l)
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 1e+275:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	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(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(l * V) <= -5e-278)
		tmp = Float64(Float64(Float64(sqrt(Float64(-A)) * c0) / sqrt(Float64(-V))) / sqrt(l));
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 1e+275)
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	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(((l / A) * V));
	tmp = 0.0;
	if ((l * V) <= -5e-278)
		tmp = ((sqrt(-A) * c0) / sqrt(-V)) / sqrt(l);
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 1e+275)
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	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[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -5e-278], N[(N[(N[(N[Sqrt[(-A)], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 1e+275], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999985e-278

   1. Initial program 76.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. associate-*r/ N/A

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

      6. frac-2neg N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. sqrt-prod N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-sqrt.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-sqrt.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-div N/A

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

      8. frac-2neg N/A

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

      9. sqrt-div N/A

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

      10. associate-*l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-neg.f64 44.7

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

   6. Applied rewrites 44.7%

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

   if -4.99999999999999985e-278 < (*.f64 V l) < 0.0 or 9.9999999999999996e274 < (*.f64 V l)

   1. Initial program 45.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. lift-sqrt.f64 N/A

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

      17. div-inv N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lift-sqrt.f64 N/A

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

   6. Applied rewrites 68.3%

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

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

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 99.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{\sqrt{-A} \cdot c0}{\sqrt{-V}}}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{-V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (/ l A) V)))))
   (if (<= (* l V) -5e-278)
     (* (/ (/ c0 (sqrt l)) (sqrt (- V))) (sqrt (- A)))
     (if (<= (* l V) 0.0)
       t_0
       (if (<= (* l V) 1e+275) (/ (* (sqrt A) c0) (sqrt (* l V))) 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(((l / A) * V));
	double tmp;
	if ((l * V) <= -5e-278) {
		tmp = ((c0 / sqrt(l)) / sqrt(-V)) * sqrt(-A);
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	} 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(((l / a) * v))
    if ((l * v) <= (-5d-278)) then
        tmp = ((c0 / sqrt(l)) / sqrt(-v)) * sqrt(-a)
    else if ((l * v) <= 0.0d0) then
        tmp = t_0
    else if ((l * v) <= 1d+275) then
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    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(((l / A) * V));
	double tmp;
	if ((l * V) <= -5e-278) {
		tmp = ((c0 / Math.sqrt(l)) / Math.sqrt(-V)) * Math.sqrt(-A);
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	} 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(((l / A) * V))
	tmp = 0
	if (l * V) <= -5e-278:
		tmp = ((c0 / math.sqrt(l)) / math.sqrt(-V)) * math.sqrt(-A)
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 1e+275:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	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(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(l * V) <= -5e-278)
		tmp = Float64(Float64(Float64(c0 / sqrt(l)) / sqrt(Float64(-V))) * sqrt(Float64(-A)));
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 1e+275)
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	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(((l / A) * V));
	tmp = 0.0;
	if ((l * V) <= -5e-278)
		tmp = ((c0 / sqrt(l)) / sqrt(-V)) * sqrt(-A);
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 1e+275)
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	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[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -5e-278], N[(N[(N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] / N[Sqrt[(-V)], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-A)], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 1e+275], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -4.99999999999999985e-278

   1. Initial program 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 76.2

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

   4. Applied rewrites 76.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. sqrt-div N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. times-frac N/A

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

      14. *-commutative N/A

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

      15. times-frac N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. associate-*l/ N/A

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

      19. associate-/r/ N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. sqrt-div N/A

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

      23. frac-2neg N/A

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

      24. sqrt-div N/A

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

   6. Applied rewrites 45.3%

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

   if -4.99999999999999985e-278 < (*.f64 V l) < 0.0 or 9.9999999999999996e274 < (*.f64 V l)

   1. Initial program 45.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 67.2

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

   4. Applied rewrites 67.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. lift-sqrt.f64 N/A

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

      17. div-inv N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lift-sqrt.f64 N/A

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

   6. Applied rewrites 68.3%

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

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

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 99.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{c0}{\sqrt{\ell}}}{\sqrt{-V}} \cdot \sqrt{-A}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (/ l A) V)))))
   (if (<= (* l V) -1e+59)
     (* (sqrt (/ A V)) (/ c0 (sqrt l)))
     (if (<= (* l V) 0.0)
       t_0
       (if (<= (* l V) 1e+275) (/ (* (sqrt A) c0) (sqrt (* l V))) 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(((l / A) * V));
	double tmp;
	if ((l * V) <= -1e+59) {
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	} 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(((l / a) * v))
    if ((l * v) <= (-1d+59)) then
        tmp = sqrt((a / v)) * (c0 / sqrt(l))
    else if ((l * v) <= 0.0d0) then
        tmp = t_0
    else if ((l * v) <= 1d+275) then
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    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(((l / A) * V));
	double tmp;
	if ((l * V) <= -1e+59) {
		tmp = Math.sqrt((A / V)) * (c0 / Math.sqrt(l));
	} else if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	} 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(((l / A) * V))
	tmp = 0
	if (l * V) <= -1e+59:
		tmp = math.sqrt((A / V)) * (c0 / math.sqrt(l))
	elif (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 1e+275:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	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(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(l * V) <= -1e+59)
		tmp = Float64(sqrt(Float64(A / V)) * Float64(c0 / sqrt(l)));
	elseif (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 1e+275)
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	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(((l / A) * V));
	tmp = 0.0;
	if ((l * V) <= -1e+59)
		tmp = sqrt((A / V)) * (c0 / sqrt(l));
	elseif ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 1e+275)
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	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[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], -1e+59], N[(N[Sqrt[N[(A / V), $MachinePrecision]], $MachinePrecision] * N[(c0 / N[Sqrt[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 1e+275], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 V l) < -9.99999999999999972e58

   1. Initial program 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sqrt-div N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 43.6

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

   4. Applied rewrites 43.6%

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

   if -9.99999999999999972e58 < (*.f64 V l) < 0.0 or 9.9999999999999996e274 < (*.f64 V l)

   1. Initial program 61.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 69.2

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

   4. Applied rewrites 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. lift-sqrt.f64 N/A

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

      17. div-inv N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lift-sqrt.f64 N/A

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

   6. Applied rewrites 70.7%

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

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

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 99.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq -1 \cdot 10^{+59}:\\ \;\;\;\;\sqrt{\frac{A}{V}} \cdot \frac{c0}{\sqrt{\ell}}\\ \mathbf{elif}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (/ l A) V)))))
   (if (<= (* l V) 0.0)
     t_0
     (if (<= (* l V) 1e+275) (/ (* (sqrt A) c0) (sqrt (* l V))) 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(((l / A) * V));
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	} 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(((l / a) * v))
    if ((l * v) <= 0.0d0) then
        tmp = t_0
    else if ((l * v) <= 1d+275) then
        tmp = (sqrt(a) * c0) / sqrt((l * v))
    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(((l / A) * V));
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (Math.sqrt(A) * c0) / Math.sqrt((l * V));
	} 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(((l / A) * V))
	tmp = 0
	if (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 1e+275:
		tmp = (math.sqrt(A) * c0) / math.sqrt((l * V))
	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(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 1e+275)
		tmp = Float64(Float64(sqrt(A) * c0) / sqrt(Float64(l * V)));
	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(((l / A) * V));
	tmp = 0.0;
	if ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 1e+275)
		tmp = (sqrt(A) * c0) / sqrt((l * V));
	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[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 1e+275], N[(N[(N[Sqrt[A], $MachinePrecision] * c0), $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.7

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

   4. Applied rewrites 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. lift-sqrt.f64 N/A

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

      17. div-inv N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lift-sqrt.f64 N/A

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

   6. Applied rewrites 73.5%

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

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

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sqrt-div N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-sqrt.f64 99.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A} \cdot c0}{\sqrt{\ell \cdot V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [c0, A, V, l] = \mathsf{sort}([c0, A, V, l])\\ \\ \begin{array}{l} t_0 := \frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (/ l A) V)))))
   (if (<= (* l V) 0.0)
     t_0
     (if (<= (* l V) 1e+275) (* (/ (sqrt A) (sqrt (* l V))) c0) 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(((l / A) * V));
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	} 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(((l / a) * v))
    if ((l * v) <= 0.0d0) then
        tmp = t_0
    else if ((l * v) <= 1d+275) then
        tmp = (sqrt(a) / sqrt((l * v))) * c0
    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(((l / A) * V));
	double tmp;
	if ((l * V) <= 0.0) {
		tmp = t_0;
	} else if ((l * V) <= 1e+275) {
		tmp = (Math.sqrt(A) / Math.sqrt((l * V))) * c0;
	} 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(((l / A) * V))
	tmp = 0
	if (l * V) <= 0.0:
		tmp = t_0
	elif (l * V) <= 1e+275:
		tmp = (math.sqrt(A) / math.sqrt((l * V))) * c0
	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(Float64(l / A) * V)))
	tmp = 0.0
	if (Float64(l * V) <= 0.0)
		tmp = t_0;
	elseif (Float64(l * V) <= 1e+275)
		tmp = Float64(Float64(sqrt(A) / sqrt(Float64(l * V))) * c0);
	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(((l / A) * V));
	tmp = 0.0;
	if ((l * V) <= 0.0)
		tmp = t_0;
	elseif ((l * V) <= 1e+275)
		tmp = (sqrt(A) / sqrt((l * V))) * c0;
	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[(N[(l / A), $MachinePrecision] * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(l * V), $MachinePrecision], 0.0], t$95$0, If[LessEqual[N[(l * V), $MachinePrecision], 1e+275], N[(N[(N[Sqrt[A], $MachinePrecision] / N[Sqrt[N[(l * V), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * c0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 72.7

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

   4. Applied rewrites 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-/r* N/A

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

      7. lift-/.f64 N/A

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

      8. sqrt-div N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. lift-/.f64 N/A

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

      14. sqrt-div N/A

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

      15. metadata-eval N/A

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

      16. lift-sqrt.f64 N/A

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

      17. div-inv N/A

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

      18. associate-/r* N/A

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

      19. lower-/.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-sqrt.f64 N/A

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

      22. lift-sqrt.f64 N/A

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

   6. Applied rewrites 73.5%

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

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

   1. Initial program 87.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 99.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot V \leq 0:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \mathbf{elif}\;\ell \cdot V \leq 10^{+275}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{\ell}{A} \cdot V}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = sqrt((a / (l * v))) * c0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Final simplification 73.7%

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

Reproduce

herbie shell --seed 2024294 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))