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

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 3

Speedup: 2.0×

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, 2.0× 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. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. sqr-abs N/A

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

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   9. metadata-eval 55.1%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

4. Applied egg-rr 55.1%

   \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]
5. Step-by-step derivation

   1. /-rgt-identity N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{x - y}{1} \cdot \left(x - y\right)\right), \frac{1}{4}\right) \]

   2. frac-2neg N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{\mathsf{neg}\left(1\right)} \cdot \left(x - y\right)\right), \frac{1}{4}\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{-1} \cdot \left(x - y\right)\right), \frac{1}{4}\right) \]

   4. remove-double-div N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{-1} \cdot \frac{1}{\frac{1}{x - y}}\right), \frac{1}{4}\right) \]

   5. frac-times N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(x - y\right)\right)\right) \cdot 1}{-1 \cdot \frac{1}{x - y}}\right), \frac{1}{4}\right) \]

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

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right) \cdot 1\right)}{-1 \cdot \frac{1}{x - y}}\right), \frac{1}{4}\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right) \cdot \frac{1}{1}\right)}{-1 \cdot \frac{1}{x - y}}\right), \frac{1}{4}\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{x - y}{1}\right)}{-1 \cdot \frac{1}{x - y}}\right), \frac{1}{4}\right) \]

   9. /-rgt-identity N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{-1 \cdot \frac{1}{x - y}}\right), \frac{1}{4}\right) \]

   10. neg-mul-1 N/A

       \[\leadsto \mathsf{pow.f64}\left(\left(\frac{\mathsf{neg}\left(\left(x - y\right)\right)}{\mathsf{neg}\left(\frac{1}{x - y}\right)}\right), \frac{1}{4}\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x - y\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x - y}\right)\right)\right), \frac{1}{4}\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x - y}\right)\right)\right), \frac{1}{4}\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x - y}\right)\right)\right), \frac{1}{4}\right) \]

   14. distribute-neg-in N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x - y}\right)\right)\right), \frac{1}{4}\right) \]

   15. remove-double-neg N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(x\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x - y}\right)\right)\right), \frac{1}{4}\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(y - x\right), \left(\mathsf{neg}\left(\frac{1}{x - y}\right)\right)\right), \frac{1}{4}\right) \]

   17. --lowering--.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\mathsf{neg}\left(\frac{1}{x - y}\right)\right)\right), \frac{1}{4}\right) \]

   18. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \left(\frac{1}{\mathsf{neg}\left(\left(x - y\right)\right)}\right)\right), \frac{1}{4}\right) \]

   19. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(x - y\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   20. sub-neg N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   21. +-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   22. distribute-neg-in N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   23. remove-double-neg N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{/.f64}\left(1, \left(y + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \frac{1}{4}\right) \]

   24. sub-neg N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{/.f64}\left(1, \left(y - x\right)\right)\right), \frac{1}{4}\right) \]

   25. --lowering--.f64 55.1%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, x\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, x\right)\right)\right), \frac{1}{4}\right) \]

6. Applied egg-rr 55.1%

   \[\leadsto {\color{blue}{\left(\frac{y - x}{\frac{1}{y - x}}\right)}}^{0.25} \]
7. Step-by-step derivation

   1. div-inv N/A

      \[\leadsto {\left(\left(y - x\right) \cdot \frac{1}{\frac{1}{y - x}}\right)}^{\frac{1}{4}} \]

   2. unpow-prod-down N/A

      \[\leadsto {\left(y - x\right)}^{\frac{1}{4}} \cdot \color{blue}{{\left(\frac{1}{\frac{1}{y - x}}\right)}^{\frac{1}{4}}} \]

   3. remove-double-div N/A

      \[\leadsto {\left(y - x\right)}^{\frac{1}{4}} \cdot {\left(y - x\right)}^{\frac{1}{4}} \]

   4. pow-prod-up N/A

      \[\leadsto {\left(y - x\right)}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}} \]

   5. metadata-eval N/A

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

   6. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(y - x\right), \color{blue}{\frac{1}{2}}\right) \]

   7. --lowering--.f64 50.0%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(y, x\right), \frac{1}{2}\right) \]

8. Applied egg-rr 50.0%

   \[\leadsto \color{blue}{{\left(y - x\right)}^{0.5}} \]
9. Step-by-step derivation

   1. unpow1/2 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(y - x\right)\right) \]

   3. --lowering--.f64 50.0%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(y, x\right)\right) \]

10. Applied egg-rr 50.0%

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

Alternative 2: 50.6% accurate, 2.0× 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. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. sqr-abs N/A

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

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   9. metadata-eval 55.1%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

4. Applied egg-rr 55.1%

   \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{pow.f64}\left(\color{blue}{\left({y}^{2}\right)}, \frac{1}{4}\right) \]
6. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(y \cdot y\right), \frac{1}{4}\right) \]

   2. *-lowering-*.f64 34.0%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{4}\right) \]

7. Simplified 34.0%

   \[\leadsto {\color{blue}{\left(y \cdot y\right)}}^{0.25} \]
8. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto {\left({y}^{2}\right)}^{\frac{1}{4}} \]

   2. pow-pow N/A

      \[\leadsto {y}^{\color{blue}{\left(2 \cdot \frac{1}{4}\right)}} \]

   3. metadata-eval N/A

      \[\leadsto {y}^{\frac{1}{2}} \]

   4. unpow1/2 N/A

      \[\leadsto \sqrt{y} \]

   5. sqrt-lowering-sqrt.f64 28.7%

      \[\leadsto \mathsf{sqrt.f64}\left(y\right) \]

9. Applied egg-rr 28.7%

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

Alternative 3: 1.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. sqr-abs N/A

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

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x - y\right), \left(x - y\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{2}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(x - y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right) \]

   9. metadata-eval 55.1%

      \[\leadsto \mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{\_.f64}\left(x, y\right)\right), \frac{1}{4}\right) \]

4. Applied egg-rr 55.1%

   \[\leadsto \color{blue}{{\left(\left(x - y\right) \cdot \left(x - y\right)\right)}^{0.25}} \]

5. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 25.5%

      \[\leadsto \mathsf{sqrt.f64}\left(x\right) \]

7. Simplified 25.5%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))