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

Percentage Accurate: 99.9% → 100.0%

Time: 4.5s

Alternatives: 6

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 100.0%

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

3. Taylor expanded in x around 0 100.0%

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

Alternative 2: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -155000000000 \lor \neg \left(x \leq 4.3 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -155000000000.0) (not (<= x 4.3e+43))) (/ x (+ y y)) 0.5))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -155000000000.0) || !(x <= 4.3e+43)) {
		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 ((x <= (-155000000000.0d0)) .or. (.not. (x <= 4.3d+43))) 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 ((x <= -155000000000.0) || !(x <= 4.3e+43)) {
		tmp = x / (y + y);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -155000000000.0) or not (x <= 4.3e+43):
		tmp = x / (y + y)
	else:
		tmp = 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -155000000000.0) || !(x <= 4.3e+43))
		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 ((x <= -155000000000.0) || ~((x <= 4.3e+43)))
		tmp = x / (y + y);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -155000000000.0], N[Not[LessEqual[x, 4.3e+43]], $MachinePrecision]], N[(x / N[(y + y), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if x < -1.55e11 or 4.3e43 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 81.3%

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

   if -1.55e11 < x < 4.3e43

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -155000000000 \lor \neg \left(x \leq 4.3 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -215000000000 \lor \neg \left(x \leq 4.2 \cdot 10^{+48}\right):\\ \;\;\;\;x \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -215000000000.0) (not (<= x 4.2e+48))) (* x (/ 0.5 y)) 0.5))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -215000000000.0) || !(x <= 4.2e+48)) {
		tmp = x * (0.5 / 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 ((x <= (-215000000000.0d0)) .or. (.not. (x <= 4.2d+48))) then
        tmp = x * (0.5d0 / y)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -215000000000.0) || !(x <= 4.2e+48)) {
		tmp = x * (0.5 / y);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -215000000000.0) or not (x <= 4.2e+48):
		tmp = x * (0.5 / y)
	else:
		tmp = 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -215000000000.0) || !(x <= 4.2e+48))
		tmp = Float64(x * Float64(0.5 / y));
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -215000000000.0) || ~((x <= 4.2e+48)))
		tmp = x * (0.5 / y);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -215000000000.0], N[Not[LessEqual[x, 4.2e+48]], $MachinePrecision]], N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], 0.5]

Derivation

1. Split input into 2 regimes

2. if x < -2.15e11 or 4.1999999999999997e48 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 81.3%

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

      1. associate-*r/ 81.3%

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

      2. *-commutative 81.3%

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

      3. associate-*r/ 81.1%

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

   5. Simplified 81.1%

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

   if -2.15e11 < x < 4.1999999999999997e48

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 75.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -215000000000 \lor \neg \left(x \leq 4.2 \cdot 10^{+48}\right):\\ \;\;\;\;x \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% 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 50.4%

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

Alternative 5: 2.6% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 62.9%

      \[\leadsto \color{blue}{\log \left(e^{\frac{x + y}{y + y}}\right)} \]

   2. *-un-lft-identity 62.9%

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

   3. exp-prod 62.9%

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

   4. flip-+ 0.0%

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

   5. +-inverses 0.0%

      \[\leadsto \log \left({\left(e^{1}\right)}^{\left(\frac{x + y}{\frac{y \cdot y - y \cdot y}{\color{blue}{0}}}\right)}\right) \]

   6. +-inverses 0.0%

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

   7. associate-/r/ 0.0%

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

   8. pow-unpow 2.0%

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

   9. +-inverses 2.7%

      \[\leadsto \log \left({\left({\left(e^{1}\right)}^{\left(\frac{x + y}{y \cdot y - y \cdot y}\right)}\right)}^{\color{blue}{0}}\right) \]

   10. metadata-eval 2.7%

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

   11. metadata-eval 2.7%

       \[\leadsto \color{blue}{0} \]

4. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Alternative 6: 2.3% accurate, 7.0× 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%

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

3. Taylor expanded in x around 0 100.0%

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

4. Taylor expanded in x around 0 50.4%

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

6. Simplified 2.3%

   \[\leadsto \color{blue}{-1} \]
7. 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 2024145 
(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)))