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

Percentage Accurate: 99.9% → 100.0%

Time: 4.1s

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(0.5 * N[(x / y), $MachinePrecision] + 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. lower-fma.f64 N/A

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

   3. lower-/.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + y}\\ t_1 := \frac{x \cdot 0.5}{y}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y y))) (t_1 (/ (* x 0.5) y)))
   (if (<= t_0 -2000000.0) t_1 (if (<= t_0 1.0) 0.5 t_1))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double t_1 = (x * 0.5) / y;
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x + y) / (y + y)
    t_1 = (x * 0.5d0) / y
    if (t_0 <= (-2000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double t_1 = (x * 0.5) / y;
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / (y + y)
	t_1 = (x * 0.5) / y
	tmp = 0
	if t_0 <= -2000000.0:
		tmp = t_1
	elif t_0 <= 1.0:
		tmp = 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + y))
	t_1 = Float64(Float64(x * 0.5) / y)
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = 0.5;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / (y + y);
	t_1 = (x * 0.5) / y;
	tmp = 0.0;
	if (t_0 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], t$95$1, If[LessEqual[t$95$0, 1.0], 0.5, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y y)) < -2e6 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -2e6 < (/.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 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{y + y} \leq -2000000:\\ \;\;\;\;\frac{x \cdot 0.5}{y}\\ \mathbf{elif}\;\frac{x + y}{y + y} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + y}\\ t_1 := x \cdot \frac{0.5}{y}\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y y))) (t_1 (* x (/ 0.5 y))))
   (if (<= t_0 -2000000.0) t_1 (if (<= t_0 1.0) 0.5 t_1))))

C

double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double t_1 = x * (0.5 / y);
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (x + y) / (y + y)
    t_1 = x * (0.5d0 / y)
    if (t_0 <= (-2000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 1.0d0) then
        tmp = 0.5d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + y) / (y + y);
	double t_1 = x * (0.5 / y);
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = 0.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + y) / (y + y)
	t_1 = x * (0.5 / y)
	tmp = 0
	if t_0 <= -2000000.0:
		tmp = t_1
	elif t_0 <= 1.0:
		tmp = 0.5
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + y) / Float64(y + y))
	t_1 = Float64(x * Float64(0.5 / y))
	tmp = 0.0
	if (t_0 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = 0.5;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + y) / (y + y);
	t_1 = x * (0.5 / y);
	tmp = 0.0;
	if (t_0 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = 0.5;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000.0], t$95$1, If[LessEqual[t$95$0, 1.0], 0.5, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (+.f64 y y)) < -2e6 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 4.4%

      \[\leadsto \left(y + y\right) \cdot \color{blue}{x} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.7%

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

   if -2e6 < (/.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 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{y + y} \leq -2000000:\\ \;\;\;\;x \cdot \frac{0.5}{y}\\ \mathbf{elif}\;\frac{x + y}{y + y} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% 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 54.2%

   \[\leadsto \color{blue}{0.5} \]
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 2024233 
(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)))