AI.Clustering.Hierarchical.Internal:average from clustering-0.2.1, A

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x / (x + y)

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x / (x + y)

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x / (x + y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{x}{x + y} \]

2. Final simplification 100.0%

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

Alternative 2: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-65}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.55e-10) (/ x y) (if (<= y 5.3e-65) (- 1.0 (/ y x)) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.55e-10) {
		tmp = x / y;
	} else if (y <= 5.3e-65) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = x / 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 <= (-1.55d-10)) then
        tmp = x / y
    else if (y <= 5.3d-65) then
        tmp = 1.0d0 - (y / x)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.55e-10) {
		tmp = x / y;
	} else if (y <= 5.3e-65) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.55e-10:
		tmp = x / y
	elif y <= 5.3e-65:
		tmp = 1.0 - (y / x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.55e-10)
		tmp = Float64(x / y);
	elseif (y <= 5.3e-65)
		tmp = Float64(1.0 - Float64(y / x));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.55e-10)
		tmp = x / y;
	elseif (y <= 5.3e-65)
		tmp = 1.0 - (y / x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.55e-10], N[(x / y), $MachinePrecision], If[LessEqual[y, 5.3e-65], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.55000000000000008e-10 or 5.30000000000000037e-65 < y

   1. Initial program 100.0%

      \[\frac{x}{x + y} \]

   2. Taylor expanded in x around 0 80.7%

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

   if -1.55000000000000008e-10 < y < 5.30000000000000037e-65

   1. Initial program 100.0%

      \[\frac{x}{x + y} \]

   2. Taylor expanded in x around inf 88.7%

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

      1. mul-1-neg 88.7%

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

      2. unsub-neg 88.7%

         \[\leadsto \color{blue}{1 - \frac{y}{x}} \]

   4. Simplified 88.7%

      \[\leadsto \color{blue}{1 - \frac{y}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-65}:\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-8) (/ x y) (if (<= y 2.8e-66) 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-8) {
		tmp = x / y;
	} else if (y <= 2.8e-66) {
		tmp = 1.0;
	} else {
		tmp = x / 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 <= (-5d-8)) then
        tmp = x / y
    else if (y <= 2.8d-66) then
        tmp = 1.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5e-8) {
		tmp = x / y;
	} else if (y <= 2.8e-66) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e-8:
		tmp = x / y
	elif y <= 2.8e-66:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e-8)
		tmp = Float64(x / y);
	elseif (y <= 2.8e-66)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e-8)
		tmp = x / y;
	elseif (y <= 2.8e-66)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e-8], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.8e-66], 1.0, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999998e-8 or 2.8e-66 < y

   1. Initial program 100.0%

      \[\frac{x}{x + y} \]

   2. Taylor expanded in x around 0 80.7%

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

   if -4.9999999999999998e-8 < y < 2.8e-66

   1. Initial program 100.0%

      \[\frac{x}{x + y} \]

   2. Taylor expanded in x around inf 88.5%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-66}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 51.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\frac{x}{x + y} \]

2. Taylor expanded in x around inf 51.0%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 51.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023202 
(FPCore (x y)
  :name "AI.Clustering.Hierarchical.Internal:average from clustering-0.2.1, A"
  :precision binary64
  (/ x (+ x y)))