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

Percentage Accurate: 99.9% → 100.0%

Time: 2.5s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\frac{x + y}{y + y} \]
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. metadata-eval N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. fma-undefine N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. *-commutative N/A

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

   11. *-lft-identity N/A

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

   12. metadata-eval N/A

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

   13. associate-*r* N/A

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

   14. times-frac N/A

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

   15. mul-1-neg N/A

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

   16. distribute-neg-frac2 N/A

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

   17. distribute-rgt-neg-out N/A

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

   18. fmm-undef N/A

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

   19. metadata-eval N/A

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

   20. --lowering--.f64 N/A

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

   21. *-lowering-*.f64 N/A

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

   22. metadata-eval N/A

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

   23. /-lowering-/.f64 99.7%

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

5. Simplified 99.7%

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

Alternative 2: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + y}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y y))))
   (if (<= x -1e-6) t_0 (if (<= x 2.2e-18) 0.5 t_0))))

C

double code(double x, double y) {
	double t_0 = x / (y + y);
	double tmp;
	if (x <= -1e-6) {
		tmp = t_0;
	} else if (x <= 2.2e-18) {
		tmp = 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + y)
    if (x <= (-1d-6)) then
        tmp = t_0
    else if (x <= 2.2d-18) then
        tmp = 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + y);
	double tmp;
	if (x <= -1e-6) {
		tmp = t_0;
	} else if (x <= 2.2e-18) {
		tmp = 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + y)
	tmp = 0
	if x <= -1e-6:
		tmp = t_0
	elif x <= 2.2e-18:
		tmp = 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + y))
	tmp = 0.0
	if (x <= -1e-6)
		tmp = t_0;
	elseif (x <= 2.2e-18)
		tmp = 0.5;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + y);
	tmp = 0.0;
	if (x <= -1e-6)
		tmp = t_0;
	elseif (x <= 2.2e-18)
		tmp = 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e-6], t$95$0, If[LessEqual[x, 2.2e-18], 0.5, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999955e-7 or 2.1999999999999998e-18 < x

   1. Initial program 99.2%

      \[\frac{x + y}{y + y} \]
   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 79.8%

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

   if -9.99999999999999955e-7 < x < 2.1999999999999998e-18

   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 75.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\frac{y}{x}}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 0.5 (/ y x))))
   (if (<= x -2.8e-8) t_0 (if (<= x 8e-18) 0.5 t_0))))

C

double code(double x, double y) {
	double t_0 = 0.5 / (y / x);
	double tmp;
	if (x <= -2.8e-8) {
		tmp = t_0;
	} else if (x <= 8e-18) {
		tmp = 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 / (y / x)
    if (x <= (-2.8d-8)) then
        tmp = t_0
    else if (x <= 8d-18) then
        tmp = 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.5 / (y / x);
	double tmp;
	if (x <= -2.8e-8) {
		tmp = t_0;
	} else if (x <= 8e-18) {
		tmp = 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.5 / (y / x)
	tmp = 0
	if x <= -2.8e-8:
		tmp = t_0
	elif x <= 8e-18:
		tmp = 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.5 / Float64(y / x))
	tmp = 0.0
	if (x <= -2.8e-8)
		tmp = t_0;
	elseif (x <= 8e-18)
		tmp = 0.5;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.5 / (y / x);
	tmp = 0.0;
	if (x <= -2.8e-8)
		tmp = t_0;
	elseif (x <= 8e-18)
		tmp = 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.5 / N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.8e-8], t$95$0, If[LessEqual[x, 8e-18], 0.5, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7999999999999999e-8 or 8.0000000000000006e-18 < x

   1. Initial program 99.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. lft-mult-inverse N/A

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

      4. associate-*l* N/A

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. *-rgt-identity N/A

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

      11. associate-*r/ N/A

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

      12. rgt-mult-inverse N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. /-lowering-/.f64 79.7%

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

   5. Simplified 79.7%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{x}}} \]

   if -2.7999999999999999e-8 < x < 8.0000000000000006e-18

   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 75.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 50.6% 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 99.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 49.2%

   \[\leadsto \color{blue}{0.5} \]
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 2024158 
(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)))