Data.Octree.Internal:octantDistance from Octree-0.5.4.2

Percentage Accurate: 54.2% → 99.6%

Time: 3.8s

Alternatives: 2

Speedup: 24.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ \mathsf{fma}\left(x\_m, \frac{x\_m \cdot 0.5}{y\_m}, y\_m\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m) :precision binary64 (fma x_m (/ (* x_m 0.5) y_m) y_m))

C

y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m);
double code(double x_m, double y_m) {
	return fma(x_m, ((x_m * 0.5) / y_m), y_m);
}

Julia

y_m = abs(y)
x_m = abs(x)
x_m, y_m = sort([x_m, y_m])
function code(x_m, y_m)
	return fma(x_m, Float64(Float64(x_m * 0.5) / y_m), y_m)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_] := N[(x$95$m * N[(N[(x$95$m * 0.5), $MachinePrecision] / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2}}{y} + y} \]

   2. *-lft-identity N/A

      \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{1 \cdot {x}^{2}}}{y} + y \]

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*l/ N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 26.0

       \[\leadsto \mathsf{fma}\left(x, \frac{\color{blue}{x \cdot 0.5}}{y}, y\right) \]

5. Simplified 26.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x \cdot 0.5}{y}, y\right)} \]
6. Add Preprocessing

Alternative 2: 99.1% accurate, 24.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ y\_m \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_m y_m) :precision binary64 y_m)

C

y_m = fabs(y);
x_m = fabs(x);
assert(x_m < y_m);
double code(double x_m, double y_m) {
	return y_m;
}

Fortran

y_m = abs(y)
x_m = abs(x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = y_m
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
assert x_m < y_m;
public static double code(double x_m, double y_m) {
	return y_m;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
[x_m, y_m] = sort([x_m, y_m])
def code(x_m, y_m):
	return y_m

Julia

y_m = abs(y)
x_m = abs(x)
x_m, y_m = sort([x_m, y_m])
function code(x_m, y_m)
	return y_m
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$m_, y$95$m_] := y$95$m

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \sqrt{\color{blue}{{y}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f64 31.5

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

5. Simplified 31.5%

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

   1. sqrt-prod N/A

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

   2. rem-square-sqrt 26.0

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

7. Applied egg-rr 26.0%

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

Developer Target 1: 53.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -1.1236950826599826 \cdot 10^{+145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x < 1.116557621183362 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (< x -1.1236950826599826e+145)
   (- x)
   (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x)))

C

double code(double x, double y) {
	double tmp;
	if (x < -1.1236950826599826e+145) {
		tmp = -x;
	} else if (x < 1.116557621183362e+93) {
		tmp = sqrt(((x * x) + (y * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x < (-1.1236950826599826d+145)) then
        tmp = -x
    else if (x < 1.116557621183362d+93) then
        tmp = sqrt(((x * x) + (y * y)))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x < -1.1236950826599826e+145) {
		tmp = -x;
	} else if (x < 1.116557621183362e+93) {
		tmp = Math.sqrt(((x * x) + (y * y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x < -1.1236950826599826e+145:
		tmp = -x
	elif x < 1.116557621183362e+93:
		tmp = math.sqrt(((x * x) + (y * y)))
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x < -1.1236950826599826e+145)
		tmp = Float64(-x);
	elseif (x < 1.116557621183362e+93)
		tmp = sqrt(Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x < -1.1236950826599826e+145)
		tmp = -x;
	elseif (x < 1.116557621183362e+93)
		tmp = sqrt(((x * x) + (y * y)));
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Less[x, -1.1236950826599826e+145], (-x), If[Less[x, 1.116557621183362e+93], N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], x]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -11236950826599826000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x) (if (< x 1116557621183362000000000000000000000000000000000000000000000000000000000000000000000000000000) (sqrt (+ (* x x) (* y y))) x)))

  (sqrt (+ (* x x) (* y y))))