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

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \sqrt{\left(1 + \frac{x}{y}\right) \cdot y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (sqrt (* (+ 1.0 (/ x y)) y)))

C

assert(x < y);
double code(double x, double y) {
	return sqrt(((1.0 + (x / y)) * 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(((1.0d0 + (x / y)) * y))
end function

Java

assert x < y;
public static double code(double x, double y) {
	return Math.sqrt(((1.0 + (x / y)) * y));
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return math.sqrt(((1.0 + (x / y)) * y))

Julia

x, y = sort([x, y])
function code(x, y)
	return sqrt(Float64(Float64(1.0 + Float64(x / y)) * y))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = sqrt(((1.0 + (x / y)) * y));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[Sqrt[N[(N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-neg-in N/A

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

   7. neg-mul-1 N/A

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

   8. remove-double-neg N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. neg-mul-1 N/A

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

   15. remove-double-neg N/A

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

   16. lower-fma.f64 47.7

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

4. Applied rewrites 47.7%

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

5. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{\color{blue}{\frac{x}{y} \cdot y + 1 \cdot y}} \]

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 88.7

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

7. Applied rewrites 88.7%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{y}, y, y\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 88.8%

   \[\leadsto \sqrt{\left(1 + \frac{x}{y}\right) \cdot \color{blue}{y}} \]
10. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-neg-in N/A

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

   7. neg-mul-1 N/A

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

   8. remove-double-neg N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. sub-neg N/A

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

   13. distribute-neg-in N/A

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

   14. neg-mul-1 N/A

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

   15. remove-double-neg N/A

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

   16. lower-fma.f64 47.7

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

4. Applied rewrites 47.7%

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

5. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{\color{blue}{\frac{x}{y} \cdot y + 1 \cdot y}} \]

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 88.7

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

7. Applied rewrites 88.7%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{y}, y, y\right)}} \]
8. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\sqrt{x + y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 4: 97.8% accurate, 1.3× 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%

   \[\sqrt{x + y} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 53.6

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

5. Applied rewrites 53.6%

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

Reproduce

herbie shell --seed 2024321 
(FPCore (x y)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, A"
  :precision binary64
  (sqrt (+ x y)))