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

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

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. 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. Add Preprocessing

5. Taylor expanded in y around -inf 100.0%

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

   1. fabs-neg 100.0%

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

   2. neg-mul-1 100.0%

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

   3. sub-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 52.0%

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

   6. fabs-sqr 52.0%

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

   7. rem-square-sqrt 52.0%

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

7. Simplified 52.0%

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

Alternative 2: 83.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{-115}:\\ \;\;\;\;\sqrt{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 3.6e-115) (sqrt (- x)) (sqrt y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.6e-115) {
		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 <= 3.6d-115) 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 <= 3.6e-115) {
		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 <= 3.6e-115:
		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 <= 3.6e-115)
		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 <= 3.6e-115)
		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, 3.6e-115], N[Sqrt[(-x)], $MachinePrecision], N[Sqrt[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.60000000000000009e-115

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 56.3%

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

      3. pow1/2 56.3%

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

      4. fabs-sub 56.3%

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

      5. fabs-sub 56.3%

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

      6. sqr-abs 56.3%

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

      7. unpow-prod-down 64.8%

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

      8. pow1/2 64.8%

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

      9. pow1/2 64.8%

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

      10. add-sqr-sqrt 64.8%

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

      11. add-sqr-sqrt 49.1%

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

      12. add-sqr-sqrt 11.9%

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

      13. prod-diff 11.9%

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

      14. add-sqr-sqrt 11.9%

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

      15. fmm-def 11.9%

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

      16. add-sqr-sqrt 11.9%

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

   6. Applied egg-rr 14.4%

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

   7. Taylor expanded in y around 0 33.5%

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

      1. mul-1-neg 33.5%

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

   9. Simplified 33.5%

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

   if 3.60000000000000009e-115 < y

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 49.6%

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

      3. pow1/2 49.6%

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

      4. fabs-sub 49.6%

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

      5. fabs-sub 49.6%

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

      6. sqr-abs 49.6%

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

      7. unpow-prod-down 20.6%

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

      8. pow1/2 20.6%

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

      9. pow1/2 20.6%

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

      10. add-sqr-sqrt 20.6%

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

      11. add-sqr-sqrt 20.6%

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

      12. add-sqr-sqrt 20.6%

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

      13. prod-diff 20.6%

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

      14. add-sqr-sqrt 20.6%

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

      15. fmm-def 20.6%

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

      16. add-sqr-sqrt 20.6%

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

   6. Applied egg-rr 79.0%

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

   7. Taylor expanded in y around inf 66.7%

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

Alternative 3: 51.5% 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. 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. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

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

   2. sqrt-unprod 53.8%

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

   3. pow1/2 53.8%

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

   4. fabs-sub 53.8%

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

   5. fabs-sub 53.8%

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

   6. sqr-abs 53.8%

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

   7. unpow-prod-down 48.0%

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

   8. pow1/2 48.0%

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

   9. pow1/2 48.0%

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

   10. add-sqr-sqrt 48.0%

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

   11. add-sqr-sqrt 38.3%

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

   12. add-sqr-sqrt 15.2%

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

   13. prod-diff 15.2%

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

   14. add-sqr-sqrt 15.2%

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

   15. fmm-def 15.2%

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

   16. add-sqr-sqrt 15.2%

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

6. Applied egg-rr 38.9%

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

7. Taylor expanded in y around inf 27.6%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, C"
  :precision binary64
  (sqrt (fabs (- x y))))