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

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 4

Speedup: 1.0×

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, 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]

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 42.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -2.1e-104) (sqrt (- x)) (sqrt y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-104) {
		tmp = sqrt(-x);
	} else {
		tmp = sqrt(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.1d-104)) then
        tmp = sqrt(-x)
    else
        tmp = sqrt(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-104) {
		tmp = Math.sqrt(-x);
	} else {
		tmp = Math.sqrt(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.1e-104:
		tmp = math.sqrt(-x)
	else:
		tmp = math.sqrt(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e-104)
		tmp = sqrt(Float64(-x));
	else
		tmp = sqrt(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.1e-104)
		tmp = sqrt(-x);
	else
		tmp = sqrt(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e-104], N[Sqrt[(-x)], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999999e-104

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

      1. pow1/2 100.0%

         \[\leadsto \color{blue}{{\left(\left|y - x\right|\right)}^{0.5}} \]

      2. add-cube-cbrt 98.8%

         \[\leadsto {\left(\left|\color{blue}{\left(\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right) \cdot \sqrt[3]{y - x}}\right|\right)}^{0.5} \]

      3. fabs-mul 98.8%

         \[\leadsto {\color{blue}{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right| \cdot \left|\sqrt[3]{y - x}\right|\right)}}^{0.5} \]

      4. unpow-prod-down 98.7%

         \[\leadsto \color{blue}{{\left(\left|\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

      5. pow2 98.7%

         \[\leadsto {\left(\left|\color{blue}{{\left(\sqrt[3]{y - x}\right)}^{2}}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5} \]

   5. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{{\left(\left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5} \cdot {\left(\left|\sqrt[3]{y - x}\right|\right)}^{0.5}} \]

   6. Taylor expanded in y around 0 66.8%

      \[\leadsto \color{blue}{\sqrt{\left|{\left(y - x\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|}} \]
   7. Step-by-step derivation

      1. sub-neg 66.8%

         \[\leadsto \sqrt{\left|{\color{blue}{\left(y + \left(-x\right)\right)}}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      2. neg-mul-1 66.8%

         \[\leadsto \sqrt{\left|{\left(y + \color{blue}{-1 \cdot x}\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y - x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      3. sub-neg 66.8%

         \[\leadsto \sqrt{\left|{\left(y + -1 \cdot x\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\color{blue}{\left(y + \left(-x\right)\right)}}^{0.3333333333333333}\right)}^{2}\right|} \]

      4. neg-mul-1 66.8%

         \[\leadsto \sqrt{\left|{\left(y + -1 \cdot x\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y + \color{blue}{-1 \cdot x}\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      5. neg-mul-1 66.8%

         \[\leadsto \sqrt{\left|{\left(y + \color{blue}{\left(-x\right)}\right)}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y + -1 \cdot x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      6. sub-neg 66.8%

         \[\leadsto \sqrt{\left|{\color{blue}{\left(y - x\right)}}^{0.3333333333333333}\right| \cdot \left|{\left({\left(y + -1 \cdot x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      7. metadata-eval 66.8%

         \[\leadsto \sqrt{\left|{\left(y - x\right)}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}\right| \cdot \left|{\left({\left(y + -1 \cdot x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      8. pow-sqr 66.8%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(y - x\right)}^{0.16666666666666666} \cdot {\left(y - x\right)}^{0.16666666666666666}}\right| \cdot \left|{\left({\left(y + -1 \cdot x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      9. pow-sqr 66.8%

         \[\leadsto \sqrt{\left|\color{blue}{{\left(y - x\right)}^{\left(2 \cdot 0.16666666666666666\right)}}\right| \cdot \left|{\left({\left(y + -1 \cdot x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      10. metadata-eval 66.8%

          \[\leadsto \sqrt{\left|{\left(y - x\right)}^{\color{blue}{0.3333333333333333}}\right| \cdot \left|{\left({\left(y + -1 \cdot x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

      11. unpow1/3 67.4%

          \[\leadsto \sqrt{\left|\color{blue}{\sqrt[3]{y - x}}\right| \cdot \left|{\left({\left(y + -1 \cdot x\right)}^{0.3333333333333333}\right)}^{2}\right|} \]

   8. Simplified 98.8%

      \[\leadsto \color{blue}{\sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|}} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 98.8%

         \[\leadsto \sqrt{\color{blue}{\sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|} \cdot \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|}}} \]

      2. pow1/2 98.8%

         \[\leadsto \sqrt{\color{blue}{{\left(\left|\sqrt[3]{y - x}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5}} \cdot \sqrt{\left|\sqrt[3]{y - x}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|}} \]

      3. pow1/2 98.8%

         \[\leadsto \sqrt{{\left(\left|\sqrt[3]{y - x}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5} \cdot \color{blue}{{\left(\left|\sqrt[3]{y - x}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)}^{0.5}}} \]

      4. pow-prod-down 46.5%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\left|\sqrt[3]{y - x}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right) \cdot \left(\left|\sqrt[3]{y - x}\right| \cdot \left|{\left(\sqrt[3]{y - x}\right)}^{2}\right|\right)\right)}^{0.5}}} \]

   10. Applied egg-rr 98.8%

       \[\leadsto \sqrt{\color{blue}{{\left({\left(\sqrt[3]{y - x}\right)}^{2}\right)}^{1.5}}} \]

   11. Taylor expanded in y around 0 62.2%

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

       1. neg-mul-1 62.2%

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

   13. Simplified 62.2%

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

   if -2.09999999999999999e-104 < x

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around -inf 100.0%

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

      1. fabs-neg 100.0%

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

      2. mul-1-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. fabs-sub 100.0%

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

      5. rem-square-sqrt 34.8%

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

      6. fabs-sqr 34.8%

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

      7. rem-sqrt-square 34.8%

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

      8. unpow1 34.8%

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

      9. sqr-pow 34.6%

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

      10. fabs-sqr 34.6%

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

      11. sqr-pow 34.8%

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

      12. unpow1 34.8%

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

   6. Simplified 34.8%

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

   7. Taylor expanded in x around 0 33.0%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-104}:\\ \;\;\;\;\sqrt{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \]

Alternative 3: 50.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. fabs-sub 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. mul-1-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. fabs-sub 100.0%

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

   5. rem-square-sqrt 50.0%

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

   6. fabs-sqr 50.0%

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

   7. rem-sqrt-square 50.0%

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

   8. unpow1 50.0%

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

   9. sqr-pow 49.7%

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

   10. fabs-sqr 49.7%

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

   11. sqr-pow 50.0%

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

   12. unpow1 50.0%

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

6. Simplified 50.0%

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

7. Final simplification 50.0%

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

Alternative 4: 26.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return sqrt(y)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Sqrt[y], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. fabs-sub 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in y around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. mul-1-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. fabs-sub 100.0%

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

   5. rem-square-sqrt 50.0%

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

   6. fabs-sqr 50.0%

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

   7. rem-sqrt-square 50.0%

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

   8. unpow1 50.0%

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

   9. sqr-pow 49.7%

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

   10. fabs-sqr 49.7%

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

   11. sqr-pow 50.0%

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

   12. unpow1 50.0%

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

6. Simplified 50.0%

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

7. Taylor expanded in x around 0 25.5%

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

8. Final simplification 25.5%

   \[\leadsto \sqrt{y} \]

Reproduce

herbie shell --seed 2023310 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))