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

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := 2 + t_0\\ \frac{t_0 \cdot t_1}{t_1} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (+ 2.0 t_0))) (/ (* t_0 t_1) t_1)))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = 2.0 + t_0;
	return (t_0 * t_1) / t_1;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    t_0 = x / (x + y)
    t_1 = 2.0d0 + t_0
    code = (t_0 * t_1) / t_1
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = 2.0 + t_0;
	return (t_0 * t_1) / t_1;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	t_1 = 2.0 + t_0
	return (t_0 * t_1) / t_1

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(2.0 + t_0)
	return Float64(Float64(t_0 * t_1) / t_1)
end

MATLAB

function tmp = code(x, y)
	t_0 = x / (x + y);
	t_1 = 2.0 + t_0;
	tmp = (t_0 * t_1) / t_1;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, N[(N[(t$95$0 * t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

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

   1. expm1-log1p-u 100.0%

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

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{x + y}\right)\right)} \]
4. Step-by-step derivation

   1. expm1-udef 60.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} - 1} \]

   2. flip-- 60.4%

      \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} \cdot e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1}} \]

   3. metadata-eval 60.4%

      \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} \cdot e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} - \color{blue}{1}}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1} \]

   4. fma-neg 60.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)}, e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)}, -1\right)}}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1} \]

   5. log1p-udef 60.4%

      \[\leadsto \frac{\mathsf{fma}\left(e^{\color{blue}{\log \left(1 + \frac{x}{x + y}\right)}}, e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)}, -1\right)}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1} \]

   6. add-exp-log 60.4%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{1 + \frac{x}{x + y}}, e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)}, -1\right)}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1} \]

   7. +-commutative 60.4%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{x + y} + 1}, e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)}, -1\right)}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1} \]

   8. log1p-udef 60.4%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{x + y} + 1, e^{\color{blue}{\log \left(1 + \frac{x}{x + y}\right)}}, -1\right)}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1} \]

   9. add-exp-log 60.4%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{x + y} + 1, \color{blue}{1 + \frac{x}{x + y}}, -1\right)}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1} \]

   10. +-commutative 60.4%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{x + y} + 1, \color{blue}{\frac{x}{x + y} + 1}, -1\right)}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1} \]

   11. metadata-eval 60.4%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{x}{x + y} + 1, \frac{x}{x + y} + 1, \color{blue}{-1}\right)}{e^{\mathsf{log1p}\left(\frac{x}{x + y}\right)} + 1} \]

   12. log1p-udef 60.4%

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

   13. add-exp-log 60.4%

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

   14. +-commutative 60.4%

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

5. Applied egg-rr 60.4%

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

   1. fma-udef 60.4%

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

   2. difference-of-sqr--1 60.4%

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

   3. +-commutative 60.4%

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

   4. +-commutative 60.4%

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

   5. associate-+r+ 60.4%

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

   6. metadata-eval 60.4%

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

   7. associate--l+ 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto \frac{\left(2 + \frac{x}{x + y}\right) \cdot \left(\frac{x}{x + y} + \color{blue}{0}\right)}{\left(\frac{x}{x + y} + 1\right) + 1} \]

   9. +-rgt-identity 100.0%

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

   10. +-commutative 100.0%

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

   11. +-commutative 100.0%

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

   12. associate-+r+ 100.0%

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

   13. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.8e-8) (/ x y) (if (<= y 4.6e-15) 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.8e-8) {
		tmp = x / y;
	} else if (y <= 4.6e-15) {
		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 <= (-2.8d-8)) then
        tmp = x / y
    else if (y <= 4.6d-15) 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 <= -2.8e-8) {
		tmp = x / y;
	} else if (y <= 4.6e-15) {
		tmp = 1.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.8e-8:
		tmp = x / y
	elif y <= 4.6e-15:
		tmp = 1.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.8e-8)
		tmp = Float64(x / y);
	elseif (y <= 4.6e-15)
		tmp = 1.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.8e-8)
		tmp = x / y;
	elseif (y <= 4.6e-15)
		tmp = 1.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.8e-8], N[(x / y), $MachinePrecision], If[LessEqual[y, 4.6e-15], 1.0, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7999999999999999e-8 or 4.59999999999999981e-15 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 80.0%

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

   if -2.7999999999999999e-8 < y < 4.59999999999999981e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 77.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 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 4: 50.3% 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 45.3%

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

3. Final simplification 45.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x y)
  :name "AI.Clustering.Hierarchical.Internal:average from clustering-0.2.1, A"
  :precision binary64
  (/ x (+ x y)))