Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2

Percentage Accurate: 99.9% → 100.0%

Time: 2.3s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \frac{\frac{x}{2}}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 0.5 (/ (/ x 2.0) y)))

C

double code(double x, double y) {
	return 0.5 + ((x / 2.0) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 + ((x / 2.0d0) / y)
end function

Java

public static double code(double x, double y) {
	return 0.5 + ((x / 2.0) / y);
}

Python

def code(x, y):
	return 0.5 + ((x / 2.0) / y)

Julia

function code(x, y)
	return Float64(0.5 + Float64(Float64(x / 2.0) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 + ((x / 2.0) / y);
end

Wolfram

code[x_, y_] := N[(0.5 + N[(N[(x / 2.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{\frac{1}{2} \cdot 1}{\color{blue}{y}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x \cdot \frac{1}{2}}{\color{blue}{y}}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{y \cdot \color{blue}{1}}\right)\right) \]

   11. lft-mult-inverse N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{y \cdot \left(\frac{1}{x} \cdot \color{blue}{x}\right)}\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\left(y \cdot \frac{1}{x}\right) \cdot \color{blue}{x}}\right)\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\frac{y \cdot 1}{x} \cdot x}\right)\right) \]

   14. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\frac{y}{x} \cdot x}\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{x \cdot \color{blue}{\frac{y}{x}}}\right)\right) \]

   16. associate-/r* N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{\frac{1}{2} \cdot x}{x}}{\color{blue}{\frac{y}{x}}}\right)\right) \]

   17. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot \frac{x}{x}}{\frac{\color{blue}{y}}{x}}\right)\right) \]

   18. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot \frac{x \cdot 1}{x}}{\frac{y}{x}}\right)\right) \]

   19. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot \left(x \cdot \frac{1}{x}\right)}{\frac{y}{x}}\right)\right) \]

   20. rgt-mult-inverse N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\frac{y}{x}}\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\frac{\color{blue}{y}}{x}}\right)\right) \]

   22. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{y}{x}\right)}\right)\right) \]

   23. /-lowering-/.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{0.5 + \frac{0.5}{\frac{y}{x}}} \]
6. Step-by-step derivation

   1. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{y} \cdot \color{blue}{x}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{y} \cdot x\right)\right) \]

   3. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2 \cdot y} \cdot x\right)\right) \]

   4. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{y + y} \cdot x\right)\right) \]

   5. associate-/r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{\color{blue}{\frac{y + y}{x}}}\right)\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\color{blue}{y + y}}\right)\right) \]

   7. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{2 \cdot \color{blue}{y}}\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{x}{2}}{\color{blue}{y}}\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{x}{2}\right), \color{blue}{y}\right)\right) \]

   10. /-lowering-/.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, 2\right), y\right)\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto 0.5 + \color{blue}{\frac{\frac{x}{2}}{y}} \]
8. Add Preprocessing

Alternative 2: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+24}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e+24) 0.5 (if (<= y 1.1e+32) (/ x (+ y y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e+24) {
		tmp = 0.5;
	} else if (y <= 1.1e+32) {
		tmp = x / (y + y);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4d+24)) then
        tmp = 0.5d0
    else if (y <= 1.1d+32) then
        tmp = x / (y + y)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4e+24) {
		tmp = 0.5;
	} else if (y <= 1.1e+32) {
		tmp = x / (y + y);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e+24:
		tmp = 0.5
	elif y <= 1.1e+32:
		tmp = x / (y + y)
	else:
		tmp = 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e+24)
		tmp = 0.5;
	elseif (y <= 1.1e+32)
		tmp = Float64(x / Float64(y + y));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4e+24)
		tmp = 0.5;
	elseif (y <= 1.1e+32)
		tmp = x / (y + y);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e+24], 0.5, If[LessEqual[y, 1.1e+32], N[(x / N[(y + y), $MachinePrecision]), $MachinePrecision], 0.5]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9999999999999999e24 or 1.1e32 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \]
   4. Step-by-step derivation

   5. Simplified 80.3%

      \[\leadsto \color{blue}{0.5} \]

   if -3.9999999999999999e24 < y < 1.1e32

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{x}, \mathsf{+.f64}\left(y, y\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 76.9%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + \frac{0.5}{\frac{y}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 0.5 (/ 0.5 (/ y x))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0 + (0.5d0 / (y / x))
end function

Java

public static double code(double x, double y) {
	return 0.5 + (0.5 / (y / x));
}

Python

def code(x, y):
	return 0.5 + (0.5 / (y / x))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 0.5 + (0.5 / (y / x));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. associate-*l/ N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{y}\right)\right)}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{\frac{1}{2} \cdot 1}{\color{blue}{y}}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{\frac{1}{2}}{y}\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x \cdot \frac{1}{2}}{\color{blue}{y}}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{y}\right)\right) \]

   10. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{y \cdot \color{blue}{1}}\right)\right) \]

   11. lft-mult-inverse N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{y \cdot \left(\frac{1}{x} \cdot \color{blue}{x}\right)}\right)\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\left(y \cdot \frac{1}{x}\right) \cdot \color{blue}{x}}\right)\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\frac{y \cdot 1}{x} \cdot x}\right)\right) \]

   14. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\frac{y}{x} \cdot x}\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{x \cdot \color{blue}{\frac{y}{x}}}\right)\right) \]

   16. associate-/r* N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{\frac{1}{2} \cdot x}{x}}{\color{blue}{\frac{y}{x}}}\right)\right) \]

   17. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot \frac{x}{x}}{\frac{\color{blue}{y}}{x}}\right)\right) \]

   18. *-rgt-identity N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot \frac{x \cdot 1}{x}}{\frac{y}{x}}\right)\right) \]

   19. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot \left(x \cdot \frac{1}{x}\right)}{\frac{y}{x}}\right)\right) \]

   20. rgt-mult-inverse N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\frac{y}{x}}\right)\right) \]

   21. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\frac{\color{blue}{y}}{x}}\right)\right) \]

   22. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{y}{x}\right)}\right)\right) \]

   23. /-lowering-/.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{0.5 + \frac{0.5}{\frac{y}{x}}} \]
6. Add Preprocessing

Alternative 4: 50.0% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.5)

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0
end function

Java

public static double code(double x, double y) {
	return 0.5;
}

Python

def code(x, y):
	return 0.5

Julia

function code(x, y)
	return 0.5
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5;
end

Wolfram

code[x_, y_] := 0.5

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Simplified 50.3%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{x}{y} + 0.5 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.5d0 * (x / y)) + 0.5d0
end function

Java

public static double code(double x, double y) {
	return (0.5 * (x / y)) + 0.5;
}

Python

def code(x, y):
	return (0.5 * (x / y)) + 0.5

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (0.5 * (x / y)) + 0.5;
end

Wolfram

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x y)
  :name "Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* 1/2 (/ x y)) 1/2))

  (/ (+ x y) (+ y y)))