Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 4

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\left|x - y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (fabs (- x y))))

C

double code(double x, double y) {
	return sqrt(fabs((x - y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(abs((x - y)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(Math.abs((x - y)));
}

Python

def code(x, y):
	return math.sqrt(math.fabs((x - y)))

Julia

function code(x, y)
	return sqrt(abs(Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(abs((x - y)));
end

Wolfram

code[x_, y_] := N[Sqrt[N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left|x - y\right|} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (fabs (- x y))))

C

double code(double x, double y) {
	return sqrt(fabs((x - y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(abs((x - y)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(Math.abs((x - y)));
}

Python

def code(x, y):
	return math.sqrt(math.fabs((x - y)))

Julia

function code(x, y)
	return sqrt(abs(Float64(x - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(abs((x - y)));
end

Wolfram

code[x_, y_] := N[Sqrt[N[Abs[N[(x - y), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \sqrt{y - x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (sqrt (- y x)))

C

assert(x < y);
double code(double x, double y) {
	return sqrt((y - x));
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((y - x))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return Math.sqrt((y - x));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return math.sqrt((y - x))

Julia

x, y = sort([x, y])
function code(x, y)
	return sqrt(Float64(y - x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = sqrt((y - x));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[Sqrt[N[(y - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{\left|x - y\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\left|x - y\right|}} \]

   2. lift-fabs.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left|x - y\right|}} \]

   3. lift--.f64 N/A

      \[\leadsto \sqrt{\left|\color{blue}{x - y}\right|} \]

   4. flip-- N/A

      \[\leadsto \sqrt{\left|\color{blue}{\frac{x \cdot x - y \cdot y}{x + y}}\right|} \]

   5. fabs-div N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left|x \cdot x - y \cdot y\right|}{\left|x + y\right|}}} \]

   6. clear-num N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

   7. sqrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}} \]

   9. div-fabs N/A

      \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

   12. div-fabs N/A

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

   13. neg-fabs N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{\left|\mathsf{neg}\left(\left(x + y\right)\right)\right|}}{\left|x \cdot x - y \cdot y\right|}}} \]

   14. div-fabs N/A

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

   15. lower-fabs.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

   16. clear-num N/A

       \[\leadsto \frac{1}{\sqrt{\left|\color{blue}{\frac{1}{\frac{x \cdot x - y \cdot y}{\mathsf{neg}\left(\left(x + y\right)\right)}}}\right|}} \]

   17. distribute-neg-frac2 N/A

       \[\leadsto \frac{1}{\sqrt{\left|\frac{1}{\color{blue}{\mathsf{neg}\left(\frac{x \cdot x - y \cdot y}{x + y}\right)}}\right|}} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{1}{y - x}\right|}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{1}{y - x}\right|}}} \]

   2. inv-pow N/A

      \[\leadsto \color{blue}{{\left(\sqrt{\left|\frac{1}{y - x}\right|}\right)}^{-1}} \]

   3. lift-sqrt.f64 N/A

      \[\leadsto {\color{blue}{\left(\sqrt{\left|\frac{1}{y - x}\right|}\right)}}^{-1} \]

   4. pow1/2 N/A

      \[\leadsto {\color{blue}{\left({\left(\left|\frac{1}{y - x}\right|\right)}^{\frac{1}{2}}\right)}}^{-1} \]

   5. pow-pow N/A

      \[\leadsto \color{blue}{{\left(\left|\frac{1}{y - x}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)}} \]

   6. lift-fabs.f64 N/A

      \[\leadsto {\color{blue}{\left(\left|\frac{1}{y - x}\right|\right)}}^{\left(\frac{1}{2} \cdot -1\right)} \]

   7. lift-/.f64 N/A

      \[\leadsto {\left(\left|\color{blue}{\frac{1}{y - x}}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)} \]

   8. inv-pow N/A

      \[\leadsto {\left(\left|\color{blue}{{\left(y - x\right)}^{-1}}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)} \]

   9. sqr-pow N/A

      \[\leadsto {\left(\left|\color{blue}{{\left(y - x\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(y - x\right)}^{\left(\frac{-1}{2}\right)}}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)} \]

   10. fabs-sqr N/A

       \[\leadsto {\color{blue}{\left({\left(y - x\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(y - x\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{\left(\frac{1}{2} \cdot -1\right)} \]

   11. sqr-pow N/A

       \[\leadsto {\color{blue}{\left({\left(y - x\right)}^{-1}\right)}}^{\left(\frac{1}{2} \cdot -1\right)} \]

   12. pow-pow N/A

       \[\leadsto \color{blue}{{\left(y - x\right)}^{\left(-1 \cdot \left(\frac{1}{2} \cdot -1\right)\right)}} \]

   13. metadata-eval N/A

       \[\leadsto {\left(y - x\right)}^{\left(-1 \cdot \color{blue}{\frac{-1}{2}}\right)} \]

   14. metadata-eval N/A

       \[\leadsto {\left(y - x\right)}^{\color{blue}{\frac{1}{2}}} \]

   15. pow1/2 N/A

       \[\leadsto \color{blue}{\sqrt{y - x}} \]

   16. lower-sqrt.f64 50.8

       \[\leadsto \color{blue}{\sqrt{y - x}} \]

6. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\sqrt{y - x}} \]
7. Add Preprocessing

Alternative 2: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|y\right|}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-102) (sqrt (- x)) (sqrt (fabs y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-102) {
		tmp = sqrt(-x);
	} else {
		tmp = sqrt(fabs(y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.5d-102)) then
        tmp = sqrt(-x)
    else
        tmp = sqrt(abs(y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-102) {
		tmp = Math.sqrt(-x);
	} else {
		tmp = Math.sqrt(Math.abs(y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9.5e-102:
		tmp = math.sqrt(-x)
	else:
		tmp = math.sqrt(math.fabs(y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-102)
		tmp = sqrt(Float64(-x));
	else
		tmp = sqrt(abs(y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e-102)
		tmp = sqrt(-x);
	else
		tmp = sqrt(abs(y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9.5e-102], N[Sqrt[(-x)], $MachinePrecision], N[Sqrt[N[Abs[y], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.50000000000000025e-102

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left|x - y\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left|x - y\right|}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left|\color{blue}{x - y}\right|} \]

      4. flip-- N/A

         \[\leadsto \sqrt{\left|\color{blue}{\frac{x \cdot x - y \cdot y}{x + y}}\right|} \]

      5. fabs-div N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left|x \cdot x - y \cdot y\right|}{\left|x + y\right|}}} \]

      6. clear-num N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

      7. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}} \]

      9. div-fabs N/A

         \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

      12. div-fabs N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

      13. neg-fabs N/A

          \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{\left|\mathsf{neg}\left(\left(x + y\right)\right)\right|}}{\left|x \cdot x - y \cdot y\right|}}} \]

      14. div-fabs N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

      15. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

      16. clear-num N/A

          \[\leadsto \frac{1}{\sqrt{\left|\color{blue}{\frac{1}{\frac{x \cdot x - y \cdot y}{\mathsf{neg}\left(\left(x + y\right)\right)}}}\right|}} \]

      17. distribute-neg-frac2 N/A

          \[\leadsto \frac{1}{\sqrt{\left|\frac{1}{\color{blue}{\mathsf{neg}\left(\frac{x \cdot x - y \cdot y}{x + y}\right)}}\right|}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{1}{y - x}\right|}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{1}{y - x}\right|}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\sqrt{\left|\frac{1}{y - x}\right|}\right)}^{-1}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{\left|\frac{1}{y - x}\right|}\right)}}^{-1} \]

      4. pow1/2 N/A

         \[\leadsto {\color{blue}{\left({\left(\left|\frac{1}{y - x}\right|\right)}^{\frac{1}{2}}\right)}}^{-1} \]

      5. pow-pow N/A

         \[\leadsto \color{blue}{{\left(\left|\frac{1}{y - x}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)}} \]

      6. lift-fabs.f64 N/A

         \[\leadsto {\color{blue}{\left(\left|\frac{1}{y - x}\right|\right)}}^{\left(\frac{1}{2} \cdot -1\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto {\left(\left|\color{blue}{\frac{1}{y - x}}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)} \]

      8. inv-pow N/A

         \[\leadsto {\left(\left|\color{blue}{{\left(y - x\right)}^{-1}}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)} \]

      9. sqr-pow N/A

         \[\leadsto {\left(\left|\color{blue}{{\left(y - x\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(y - x\right)}^{\left(\frac{-1}{2}\right)}}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)} \]

      10. fabs-sqr N/A

          \[\leadsto {\color{blue}{\left({\left(y - x\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(y - x\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{\left(\frac{1}{2} \cdot -1\right)} \]

      11. sqr-pow N/A

          \[\leadsto {\color{blue}{\left({\left(y - x\right)}^{-1}\right)}}^{\left(\frac{1}{2} \cdot -1\right)} \]

      12. pow-pow N/A

          \[\leadsto \color{blue}{{\left(y - x\right)}^{\left(-1 \cdot \left(\frac{1}{2} \cdot -1\right)\right)}} \]

      13. metadata-eval N/A

          \[\leadsto {\left(y - x\right)}^{\left(-1 \cdot \color{blue}{\frac{-1}{2}}\right)} \]

      14. metadata-eval N/A

          \[\leadsto {\left(y - x\right)}^{\color{blue}{\frac{1}{2}}} \]

      15. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{y - x}} \]

      16. lower-sqrt.f64 88.2

          \[\leadsto \color{blue}{\sqrt{y - x}} \]

   6. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\sqrt{y - x}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \sqrt{\color{blue}{-1 \cdot x}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      2. lower-neg.f64 72.9

         \[\leadsto \sqrt{\color{blue}{-x}} \]

   9. Applied rewrites 72.9%

      \[\leadsto \sqrt{\color{blue}{-x}} \]

   if -9.50000000000000025e-102 < x

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \sqrt{\left|\color{blue}{-1 \cdot y}\right|} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left|\color{blue}{\mathsf{neg}\left(y\right)}\right|} \]

      2. lower-neg.f64 61.2

         \[\leadsto \sqrt{\left|\color{blue}{-y}\right|} \]

   5. Applied rewrites 61.2%

      \[\leadsto \sqrt{\left|\color{blue}{-y}\right|} \]
   6. Step-by-step derivation

   7. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\sqrt{\left|y\right|}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 6.8e-121) (sqrt (- x)) (sqrt y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-121) {
		tmp = sqrt(-x);
	} else {
		tmp = sqrt(y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.8d-121) then
        tmp = sqrt(-x)
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.8e-121) {
		tmp = Math.sqrt(-x);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 6.8e-121:
		tmp = math.sqrt(-x)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 6.8e-121)
		tmp = sqrt(Float64(-x));
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.8e-121)
		tmp = sqrt(-x);
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 6.8e-121], N[Sqrt[(-x)], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.80000000000000003e-121

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left|x - y\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left|x - y\right|}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left|\color{blue}{x - y}\right|} \]

      4. flip-- N/A

         \[\leadsto \sqrt{\left|\color{blue}{\frac{x \cdot x - y \cdot y}{x + y}}\right|} \]

      5. fabs-div N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left|x \cdot x - y \cdot y\right|}{\left|x + y\right|}}} \]

      6. clear-num N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

      7. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}} \]

      9. div-fabs N/A

         \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

      12. div-fabs N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

      13. neg-fabs N/A

          \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{\left|\mathsf{neg}\left(\left(x + y\right)\right)\right|}}{\left|x \cdot x - y \cdot y\right|}}} \]

      14. div-fabs N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

      15. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

      16. clear-num N/A

          \[\leadsto \frac{1}{\sqrt{\left|\color{blue}{\frac{1}{\frac{x \cdot x - y \cdot y}{\mathsf{neg}\left(\left(x + y\right)\right)}}}\right|}} \]

      17. distribute-neg-frac2 N/A

          \[\leadsto \frac{1}{\sqrt{\left|\frac{1}{\color{blue}{\mathsf{neg}\left(\frac{x \cdot x - y \cdot y}{x + y}\right)}}\right|}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{1}{y - x}\right|}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{1}{y - x}\right|}}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{\left(\sqrt{\left|\frac{1}{y - x}\right|}\right)}^{-1}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{\left|\frac{1}{y - x}\right|}\right)}}^{-1} \]

      4. pow1/2 N/A

         \[\leadsto {\color{blue}{\left({\left(\left|\frac{1}{y - x}\right|\right)}^{\frac{1}{2}}\right)}}^{-1} \]

      5. pow-pow N/A

         \[\leadsto \color{blue}{{\left(\left|\frac{1}{y - x}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)}} \]

      6. lift-fabs.f64 N/A

         \[\leadsto {\color{blue}{\left(\left|\frac{1}{y - x}\right|\right)}}^{\left(\frac{1}{2} \cdot -1\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto {\left(\left|\color{blue}{\frac{1}{y - x}}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)} \]

      8. inv-pow N/A

         \[\leadsto {\left(\left|\color{blue}{{\left(y - x\right)}^{-1}}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)} \]

      9. sqr-pow N/A

         \[\leadsto {\left(\left|\color{blue}{{\left(y - x\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(y - x\right)}^{\left(\frac{-1}{2}\right)}}\right|\right)}^{\left(\frac{1}{2} \cdot -1\right)} \]

      10. fabs-sqr N/A

          \[\leadsto {\color{blue}{\left({\left(y - x\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(y - x\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{\left(\frac{1}{2} \cdot -1\right)} \]

      11. sqr-pow N/A

          \[\leadsto {\color{blue}{\left({\left(y - x\right)}^{-1}\right)}}^{\left(\frac{1}{2} \cdot -1\right)} \]

      12. pow-pow N/A

          \[\leadsto \color{blue}{{\left(y - x\right)}^{\left(-1 \cdot \left(\frac{1}{2} \cdot -1\right)\right)}} \]

      13. metadata-eval N/A

          \[\leadsto {\left(y - x\right)}^{\left(-1 \cdot \color{blue}{\frac{-1}{2}}\right)} \]

      14. metadata-eval N/A

          \[\leadsto {\left(y - x\right)}^{\color{blue}{\frac{1}{2}}} \]

      15. pow1/2 N/A

          \[\leadsto \color{blue}{\sqrt{y - x}} \]

      16. lower-sqrt.f64 29.9

          \[\leadsto \color{blue}{\sqrt{y - x}} \]

   6. Applied rewrites 29.9%

      \[\leadsto \color{blue}{\sqrt{y - x}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \sqrt{\color{blue}{-1 \cdot x}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      2. lower-neg.f64 28.5

         \[\leadsto \sqrt{\color{blue}{-x}} \]

   9. Applied rewrites 28.5%

      \[\leadsto \sqrt{\color{blue}{-x}} \]

   if 6.80000000000000003e-121 < y

   1. Initial program 100.0%

      \[\sqrt{\left|x - y\right|} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left|x - y\right|}} \]

      2. lift-fabs.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left|x - y\right|}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left|\color{blue}{x - y}\right|} \]

      4. flip-- N/A

         \[\leadsto \sqrt{\left|\color{blue}{\frac{x \cdot x - y \cdot y}{x + y}}\right|} \]

      5. fabs-div N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\left|x \cdot x - y \cdot y\right|}{\left|x + y\right|}}} \]

      6. clear-num N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

      7. sqrt-div N/A

         \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}} \]

      9. div-fabs N/A

         \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

      12. div-fabs N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

      13. neg-fabs N/A

          \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{\left|\mathsf{neg}\left(\left(x + y\right)\right)\right|}}{\left|x \cdot x - y \cdot y\right|}}} \]

      14. div-fabs N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

      15. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

      16. clear-num N/A

          \[\leadsto \frac{1}{\sqrt{\left|\color{blue}{\frac{1}{\frac{x \cdot x - y \cdot y}{\mathsf{neg}\left(\left(x + y\right)\right)}}}\right|}} \]

      17. distribute-neg-frac2 N/A

          \[\leadsto \frac{1}{\sqrt{\left|\frac{1}{\color{blue}{\mathsf{neg}\left(\frac{x \cdot x - y \cdot y}{x + y}\right)}}\right|}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{1}{y - x}\right|}}} \]

   6. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{y - x}{\sqrt{y - x}}} \]

   7. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\sqrt{y}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 63.5

         \[\leadsto \color{blue}{\sqrt{y}} \]

   9. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\sqrt{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 49.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \sqrt{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (sqrt y))

C

assert(x < y);
double code(double x, double y) {
	return sqrt(y);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(y)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return Math.sqrt(y);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return math.sqrt(y)

Julia

x, y = sort([x, y])
function code(x, y)
	return sqrt(y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = sqrt(y);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[Sqrt[y], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\sqrt{\left|x - y\right|} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\left|x - y\right|}} \]

   2. lift-fabs.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left|x - y\right|}} \]

   3. lift--.f64 N/A

      \[\leadsto \sqrt{\left|\color{blue}{x - y}\right|} \]

   4. flip-- N/A

      \[\leadsto \sqrt{\left|\color{blue}{\frac{x \cdot x - y \cdot y}{x + y}}\right|} \]

   5. fabs-div N/A

      \[\leadsto \sqrt{\color{blue}{\frac{\left|x \cdot x - y \cdot y\right|}{\left|x + y\right|}}} \]

   6. clear-num N/A

      \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

   7. sqrt-div N/A

      \[\leadsto \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\sqrt{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}} \]

   9. div-fabs N/A

      \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

   11. lower-sqrt.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\sqrt{\left|\frac{x + y}{x \cdot x - y \cdot y}\right|}}} \]

   12. div-fabs N/A

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\left|x + y\right|}{\left|x \cdot x - y \cdot y\right|}}}} \]

   13. neg-fabs N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\color{blue}{\left|\mathsf{neg}\left(\left(x + y\right)\right)\right|}}{\left|x \cdot x - y \cdot y\right|}}} \]

   14. div-fabs N/A

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

   15. lower-fabs.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\left|\frac{\mathsf{neg}\left(\left(x + y\right)\right)}{x \cdot x - y \cdot y}\right|}}} \]

   16. clear-num N/A

       \[\leadsto \frac{1}{\sqrt{\left|\color{blue}{\frac{1}{\frac{x \cdot x - y \cdot y}{\mathsf{neg}\left(\left(x + y\right)\right)}}}\right|}} \]

   17. distribute-neg-frac2 N/A

       \[\leadsto \frac{1}{\sqrt{\left|\frac{1}{\color{blue}{\mathsf{neg}\left(\frac{x \cdot x - y \cdot y}{x + y}\right)}}\right|}} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\left|\frac{1}{y - x}\right|}}} \]

6. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\frac{y - x}{\sqrt{y - x}}} \]

7. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\sqrt{y}} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 29.4

      \[\leadsto \color{blue}{\sqrt{y}} \]

9. Applied rewrites 29.4%

   \[\leadsto \color{blue}{\sqrt{y}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024243 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))