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

Percentage Accurate: 100.0% → 99.9%

Time: 3.0s

Alternatives: 5

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: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + y}\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (expm1 (log1p (/ x (+ x y)))))

C

double code(double x, double y) {
	return expm1(log1p((x / (x + y))));
}

Java

public static double code(double x, double y) {
	return Math.expm1(Math.log1p((x / (x + y))));
}

Python

def code(x, y):
	return math.expm1(math.log1p((x / (x + y))))

Julia

function code(x, y)
	return expm1(log1p(Float64(x / Float64(x + y))))
end

Wolfram

code[x_, y_] := N[(Exp[N[Log[1 + N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + y}\right)\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + y}\right)\right)} \]

5. Final simplification 100.0%

   \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + y}\right)\right) \]
6. Add Preprocessing

Alternative 2: 73.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-103} \lor \neg \left(x \leq 5 \cdot 10^{-71}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3e-103) (not (<= x 5e-71))) (- 1.0 (/ y x)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3e-103) || !(x <= 5e-71)) {
		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 ((x <= (-3d-103)) .or. (.not. (x <= 5d-71))) 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 ((x <= -3e-103) || !(x <= 5e-71)) {
		tmp = 1.0 - (y / x);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3e-103) or not (x <= 5e-71):
		tmp = 1.0 - (y / x)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3e-103) || !(x <= 5e-71))
		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 ((x <= -3e-103) || ~((x <= 5e-71)))
		tmp = 1.0 - (y / x);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3e-103], N[Not[LessEqual[x, 5e-71]], $MachinePrecision]], N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3e-103 or 4.99999999999999998e-71 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. unsub-neg 74.6%

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

   5. Simplified 74.6%

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

   if -3e-103 < x < 4.99999999999999998e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 88.4%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-103} \lor \neg \left(x \leq 5 \cdot 10^{-71}\right):\\ \;\;\;\;1 - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3e-103) 1.0 (if (<= x 2.6e-71) (/ x y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3e-103) {
		tmp = 1.0;
	} else if (x <= 2.6e-71) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3d-103)) then
        tmp = 1.0d0
    else if (x <= 2.6d-71) then
        tmp = x / y
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3e-103) {
		tmp = 1.0;
	} else if (x <= 2.6e-71) {
		tmp = x / y;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3e-103:
		tmp = 1.0
	elif x <= 2.6e-71:
		tmp = x / y
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3e-103)
		tmp = 1.0;
	elseif (x <= 2.6e-71)
		tmp = Float64(x / y);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3e-103)
		tmp = 1.0;
	elseif (x <= 2.6e-71)
		tmp = x / y;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3e-103], 1.0, If[LessEqual[x, 2.6e-71], N[(x / y), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -3e-103 or 2.5999999999999999e-71 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 74.1%

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

   if -3e-103 < x < 2.5999999999999999e-71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 88.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-103}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x}{x + y} \]
4. Add Preprocessing

Alternative 5: 51.0% 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. Add Preprocessing

3. Taylor expanded in x around inf 53.1%

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

4. Final simplification 53.1%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (x y)
  :name "AI.Clustering.Hierarchical.Internal:average from clustering-0.2.1, A"
  :precision binary64
  (/ x (+ x y)))