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

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

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, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \sqrt{\left(y + x\right) \cdot \frac{y - x}{y + x}} \end{array} \]

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return sqrt(((y + x) * ((y - x) / (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) * ((y - x) / (y + x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

      \[\leadsto \sqrt{\color{blue}{\sqrt{\left(x - y\right) \cdot \left(x - y\right)}}} \]

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 51.9

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. difference-of-squares N/A

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

   8. lift--.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. rem-square-sqrt N/A

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

   15. sqrt-prod N/A

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

   16. rem-sqrt-square-rev N/A

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

   17. lift--.f64 N/A

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

   18. fabs-sub N/A

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

   19. rem-sqrt-square-rev N/A

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

   20. sqrt-prod N/A

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

   21. rem-square-sqrt N/A

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

   22. lower--.f64 N/A

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

   23. +-commutative N/A

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

   24. lower-+.f64 48.0

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

4. Applied rewrites 48.0%

   \[\leadsto \sqrt{\color{blue}{\left(y + x\right) \cdot \frac{y - x}{y + x}}} \]
5. Add Preprocessing

Alternative 2: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.12e-117) (sqrt (- x)) (sqrt y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.12e-117) {
		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 <= 1.12d-117) 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 <= 1.12e-117) {
		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 <= 1.12e-117:
		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 <= 1.12e-117)
		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 <= 1.12e-117)
		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, 1.12e-117], N[Sqrt[(-x)], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.12e-117

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\left(x - y\right) \cdot \left(x - y\right)}}} \]

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 66.3

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. sqrt-prod N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. lift--.f64 N/A

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

      18. fabs-sub N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. rem-square-sqrt N/A

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

      22. lower--.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 33.7

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

   4. Applied rewrites 33.7%

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

   5. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 29.1

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

   7. Applied rewrites 29.1%

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

   if 1.12e-117 < y

   1. Initial program 100.0%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\left(x - y\right) \cdot \left(x - y\right)}}} \]

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 21.0

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. rem-square-sqrt N/A

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

      15. sqrt-prod N/A

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

      16. rem-sqrt-square-rev N/A

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

      17. lift--.f64 N/A

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

      18. fabs-sub N/A

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

      19. rem-sqrt-square-rev N/A

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

      20. sqrt-prod N/A

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

      21. rem-square-sqrt N/A

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

      22. lower--.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 79.0

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 72.6

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

   7. Applied rewrites 72.6%

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

Alternative 3: 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-fabs.f64 N/A

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

   2. lift--.f64 N/A

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

   3. fabs-sub N/A

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

   4. rem-sqrt-square-rev N/A

      \[\leadsto \sqrt{\color{blue}{\sqrt{\left(y - x\right) \cdot \left(y - x\right)}}} \]

   5. sqrt-prod N/A

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

   6. rem-square-sqrt N/A

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

   7. lower--.f64 48.0

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

4. Applied rewrites 48.0%

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

Alternative 4: 50.2% 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-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

      \[\leadsto \sqrt{\color{blue}{\sqrt{\left(x - y\right) \cdot \left(x - y\right)}}} \]

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 51.9

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. difference-of-squares N/A

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

   8. lift--.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. rem-square-sqrt N/A

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

   15. sqrt-prod N/A

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

   16. rem-sqrt-square-rev N/A

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

   17. lift--.f64 N/A

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

   18. fabs-sub N/A

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

   19. rem-sqrt-square-rev N/A

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

   20. sqrt-prod N/A

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

   21. rem-square-sqrt N/A

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

   22. lower--.f64 N/A

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

   23. +-commutative N/A

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

   24. lower-+.f64 48.0

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

4. Applied rewrites 48.0%

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

5. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 27.8

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

7. Applied rewrites 27.8%

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))