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

Percentage Accurate: 99.9% → 99.9%

Time: 2.8s

Alternatives: 5

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: 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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + y}\\ \mathbf{if}\;t\_0 \leq -400000000 \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;\frac{0.5 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y y))))
   (if (or (<= t_0 -400000000.0) (not (<= t_0 1.0))) (/ (* 0.5 x) y) 0.5)))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double tmp;
	if ((t_0 <= -400000000.0) || !(t_0 <= 1.0)) {
		tmp = (0.5 * x) / 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (y + y)
    if ((t_0 <= (-400000000.0d0)) .or. (.not. (t_0 <= 1.0d0))) then
        tmp = (0.5d0 * x) / y
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double tmp;
	if ((t_0 <= -400000000.0) || !(t_0 <= 1.0)) {
		tmp = (0.5 * x) / y;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / (y + y)
	tmp = 0
	if (t_0 <= -400000000.0) or not (t_0 <= 1.0):
		tmp = (0.5 * x) / y
	else:
		tmp = 0.5
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + y))
	tmp = 0.0
	if ((t_0 <= -400000000.0) || !(t_0 <= 1.0))
		tmp = Float64(Float64(0.5 * x) / y);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / (y + y);
	tmp = 0.0;
	if ((t_0 <= -400000000.0) || ~((t_0 <= 1.0)))
		tmp = (0.5 * x) / y;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -400000000.0], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[(0.5 * x), $MachinePrecision] / y), $MachinePrecision], 0.5]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y y)) < -4e8 or 1 < (/.f64 (+.f64 x y) (+.f64 y y))

   1. Initial program 100.0%

      \[\frac{x + y}{y + y} \]
   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. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 99.1%

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

   if -4e8 < (/.f64 (+.f64 x y) (+.f64 y y)) < 1

   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. Applied rewrites 97.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{y + y} \leq -400000000 \lor \neg \left(\frac{x + y}{y + y} \leq 1\right):\\ \;\;\;\;\frac{0.5 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + y}\\ \mathbf{if}\;t\_0 \leq -400000000 \lor \neg \left(t\_0 \leq 1\right):\\ \;\;\;\;\frac{0.5}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y y))))
   (if (or (<= t_0 -400000000.0) (not (<= t_0 1.0))) (* (/ 0.5 y) x) 0.5)))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double tmp;
	if ((t_0 <= -400000000.0) || !(t_0 <= 1.0)) {
		tmp = (0.5 / y) * x;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) / (y + y)
    if ((t_0 <= (-400000000.0d0)) .or. (.not. (t_0 <= 1.0d0))) then
        tmp = (0.5d0 / y) * x
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double tmp;
	if ((t_0 <= -400000000.0) || !(t_0 <= 1.0)) {
		tmp = (0.5 / y) * x;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / (y + y)
	tmp = 0
	if (t_0 <= -400000000.0) or not (t_0 <= 1.0):
		tmp = (0.5 / y) * x
	else:
		tmp = 0.5
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + y))
	tmp = 0.0
	if ((t_0 <= -400000000.0) || !(t_0 <= 1.0))
		tmp = Float64(Float64(0.5 / y) * x);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / (y + y);
	tmp = 0.0;
	if ((t_0 <= -400000000.0) || ~((t_0 <= 1.0)))
		tmp = (0.5 / y) * x;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -400000000.0], N[Not[LessEqual[t$95$0, 1.0]], $MachinePrecision]], N[(N[(0.5 / y), $MachinePrecision] * x), $MachinePrecision], 0.5]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y y)) < -4e8 or 1 < (/.f64 (+.f64 x y) (+.f64 y y))

   1. Initial program 100.0%

      \[\frac{x + y}{y + y} \]
   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. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -4e8 < (/.f64 (+.f64 x y) (+.f64 y y)) < 1

   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. Applied rewrites 97.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{y + y} \leq -400000000 \lor \neg \left(\frac{x + y}{y + y} \leq 1\right):\\ \;\;\;\;\frac{0.5}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.5}{y}, x, 0.5\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 99.9

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.5}{y}}, x, 0.5\right) \]

5. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{y}, x, 0.5\right)} \]
6. Add Preprocessing

Alternative 5: 50.0% accurate, 18.0× 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. Applied rewrites 49.6%

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

Developer Target 1: 100.0% accurate, 0.9× 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 2024313 
(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)))