Data.Octree.Internal:octantDistance from Octree-0.5.4.2

Percentage Accurate: 54.6% → 100.0%

Time: 20.4s

Alternatives: 7

Speedup: 0.2×

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: 54.6% 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: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(y, x\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision]

Derivation

1. Initial program 51.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(y, x\right)} \]
5. Add Preprocessing

Alternative 2: 48.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0 \lor \neg \left(y \cdot y \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* y y) 0.0) (not (<= (* y y) 5e+300)))
   (* (/ y x) x)
   (sqrt (fma y y (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) <= 0.0) || !((y * y) <= 5e+300)) {
		tmp = (y / x) * x;
	} else {
		tmp = sqrt(fma(y, y, (x * x)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(y * y) <= 0.0) || !(Float64(y * y) <= 5e+300))
		tmp = Float64(Float64(y / x) * x);
	else
		tmp = sqrt(fma(y, y, Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(y * y), $MachinePrecision], 0.0], N[Not[LessEqual[N[(y * y), $MachinePrecision], 5e+300]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], N[Sqrt[N[(y * y + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.0 or 5.00000000000000026e300 < (*.f64 y y)

   1. Initial program 25.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 25.5

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

   7. Applied rewrites 25.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{y} + \frac{y}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 14.1%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{y}{x} \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 18.8%

       \[\leadsto \frac{y}{x} \cdot x \]

   if 0.0 < (*.f64 y y) < 5.00000000000000026e300

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 76.6

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

   4. Applied rewrites 76.6%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0 \lor \neg \left(y \cdot y \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 35.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0 \lor \neg \left(y \cdot y \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* y y) 0.0) (not (<= (* y y) 5e+300)))
   (* (/ y x) x)
   (sqrt (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) <= 0.0) || !((y * y) <= 5e+300)) {
		tmp = (y / x) * x;
	} else {
		tmp = sqrt((y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((y * y) <= 0.0d0) .or. (.not. ((y * y) <= 5d+300))) then
        tmp = (y / x) * x
    else
        tmp = sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y * y) <= 0.0) || !((y * y) <= 5e+300)) {
		tmp = (y / x) * x;
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) <= 0.0) or not ((y * y) <= 5e+300):
		tmp = (y / x) * x
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(y * y) <= 0.0) || !(Float64(y * y) <= 5e+300))
		tmp = Float64(Float64(y / x) * x);
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * y) <= 0.0) || ~(((y * y) <= 5e+300)))
		tmp = (y / x) * x;
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(y * y), $MachinePrecision], 0.0], N[Not[LessEqual[N[(y * y), $MachinePrecision], 5e+300]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision], N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.0 or 5.00000000000000026e300 < (*.f64 y y)

   1. Initial program 25.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 25.5

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

   7. Applied rewrites 25.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{y} + \frac{y}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 14.1%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{y}{x} \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 18.8%

       \[\leadsto \frac{y}{x} \cdot x \]

   if 0.0 < (*.f64 y y) < 5.00000000000000026e300

   1. Initial program 76.6%

      \[\sqrt{x \cdot x + y \cdot y} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0 \lor \neg \left(y \cdot y \leq 5 \cdot 10^{+300}\right):\\ \;\;\;\;\frac{y}{x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 26.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e+175) (fma (/ 0.5 y) (* x x) y) (* (/ y x) x)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e+175) {
		tmp = fma((0.5 / y), (x * x), y);
	} else {
		tmp = (y / x) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e+175)
		tmp = fma(Float64(0.5 / y), Float64(x * x), y);
	else
		tmp = Float64(Float64(y / x) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+175], N[(N[(0.5 / y), $MachinePrecision] * N[(x * x), $MachinePrecision] + y), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.9999999999999999e175

   1. Initial program 68.9%

      \[\sqrt{x \cdot x + y \cdot y} \]
   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. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 37.3

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

   5. Applied rewrites 37.3%

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

   if 1.9999999999999999e175 < (*.f64 x x)

   1. Initial program 24.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-hypot.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{y} + \frac{y}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 13.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{y}{x} \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 12.7%

       \[\leadsto \frac{y}{x} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+175}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, x \cdot x, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 26.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.5}{y} \cdot x, x, y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* (/ 0.5 y) x) x y))

C

double code(double x, double y) {
	return fma(((0.5 / y) * x), x, y);
}

Julia

function code(x, y)
	return fma(Float64(Float64(0.5 / y) * x), x, y)
end

Wolfram

code[x_, y_] := N[(N[(N[(0.5 / y), $MachinePrecision] * x), $MachinePrecision] * x + y), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l* N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 27.0

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

7. Applied rewrites 27.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{y}, x \cdot x, y\right)} \]
8. Step-by-step derivation

9. Applied rewrites 28.1%

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

Alternative 6: 29.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\sqrt{x \cdot x + y \cdot y} \]
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 32.2

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

5. Applied rewrites 32.2%

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

Alternative 7: 26.6% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 28.4

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

5. Applied rewrites 28.4%

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

6. Final simplification 28.4%

   \[\leadsto -x \]
7. Add Preprocessing

Developer Target 1: 74.7% 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 2024321 
(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))))