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

Percentage Accurate: 99.9% → 100.0%

Time: 6.0s

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

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

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. accelerator-lowering-fma.f64 N/A

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

   3. /-lowering-/.f64 100.0

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

5. Simplified 100.0%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x + y}{y + y}\\ t_1 := \frac{x}{y + y}\\ \mathbf{if}\;t\_0 \leq -10000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x y) (+ y y))) (t_1 (/ x (+ y y))))
   (if (<= t_0 -10000.0) 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 / (y + y);
	double tmp;
	if (t_0 <= -10000.0) {
		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 / (y + y)
    if (t_0 <= (-10000.0d0)) 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 / (y + y);
	double tmp;
	if (t_0 <= -10000.0) {
		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 / (y + y)
	tmp = 0
	if t_0 <= -10000.0:
		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(y + y))
	tmp = 0.0
	if (t_0 <= -10000.0)
		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 / (y + y);
	tmp = 0.0;
	if (t_0 <= -10000.0)
		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[(y + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10000.0], 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)) < -1e4 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

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

   5. Simplified 97.5%

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

   if -1e4 < (/.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. Simplified 97.2%

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

Alternative 3: 50.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{+200}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y + y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 5.6e+200) 0.5 (* x (* x (+ y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.6e+200) {
		tmp = 0.5;
	} else {
		tmp = x * (x * (y + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 5.6d+200) then
        tmp = 0.5d0
    else
        tmp = x * (x * (y + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.6e+200) {
		tmp = 0.5;
	} else {
		tmp = x * (x * (y + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.6e+200:
		tmp = 0.5
	else:
		tmp = x * (x * (y + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.6e+200)
		tmp = 0.5;
	else
		tmp = Float64(x * Float64(x * Float64(y + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.6e+200)
		tmp = 0.5;
	else
		tmp = x * (x * (y + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.6e+200], 0.5, N[(x * N[(x * N[(y + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.59999999999999969e200

   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 53.5%

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

   if 5.59999999999999969e200 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 91.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. flip-+ N/A

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

      4. +-inverses N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0}}{y - y}} \cdot x \]

      5. +-inverses N/A

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

      6. +-inverses N/A

         \[\leadsto \frac{1}{\frac{y - y}{\color{blue}{0}}} \cdot x \]

      7. +-inverses N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

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

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

      11. +-lowering-+.f64 4.4

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

   7. Applied egg-rr 4.4%

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

      1. flip-+ N/A

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

      2. +-inverses N/A

         \[\leadsto \frac{\color{blue}{0}}{y - y} \cdot x \]

      3. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0 \cdot x}}{y - y} \cdot x \]

      4. +-inverses N/A

         \[\leadsto \frac{0 \cdot x}{\color{blue}{0}} \cdot x \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\left(\frac{0}{0} \cdot x\right)} \cdot x \]

      6. +-inverses N/A

         \[\leadsto \left(\frac{\color{blue}{y \cdot y - y \cdot y}}{0} \cdot x\right) \cdot x \]

      7. +-inverses N/A

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

      8. flip-+ N/A

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

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

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

      10. +-lowering-+.f64 36.6

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

   9. Applied egg-rr 36.6%

      \[\leadsto \color{blue}{\left(\left(y + y\right) \cdot x\right)} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{+200}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(y + y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 49.4% 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. Simplified 49.6%

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

Alternative 6: 2.6% accurate, 18.0× 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. Taylor expanded in x around inf

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

5. Simplified 50.8%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. flip-+ N/A

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

   4. +-inverses N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{0}}{y - y}} \cdot x \]

   5. +-inverses N/A

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

   6. +-inverses N/A

      \[\leadsto \frac{1}{\frac{y - y}{\color{blue}{0}}} \cdot x \]

   7. +-inverses N/A

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

   8. clear-num N/A

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

   9. flip-+ N/A

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

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

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

   11. +-lowering-+.f64 3.8

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

7. Applied egg-rr 3.8%

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

   1. flip-+ N/A

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

   2. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0}}{y - y} \cdot x \]

   3. mul0-lft N/A

      \[\leadsto \frac{\color{blue}{0 \cdot x}}{y - y} \cdot x \]

   4. +-inverses N/A

      \[\leadsto \frac{0 \cdot x}{\color{blue}{0}} \cdot x \]

   5. associate-*l/ N/A

      \[\leadsto \color{blue}{\left(\frac{0}{0} \cdot x\right)} \cdot x \]

   6. +-inverses N/A

      \[\leadsto \left(\frac{\color{blue}{y \cdot y - y \cdot y}}{0} \cdot x\right) \cdot x \]

   7. +-inverses N/A

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

   8. flip-+ N/A

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

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

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

   10. +-lowering-+.f64 5.6

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

9. Applied egg-rr 5.6%

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

    1. associate-*l* N/A

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

    2. flip-+ N/A

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

    3. +-inverses N/A

       \[\leadsto \frac{\color{blue}{0}}{y - y} \cdot \left(x \cdot x\right) \]

    4. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{0 - 0}}{y - y} \cdot \left(x \cdot x\right) \]

    5. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{0 \cdot 0} - 0}{y - y} \cdot \left(x \cdot x\right) \]

    6. metadata-eval N/A

       \[\leadsto \frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{y - y} \cdot \left(x \cdot x\right) \]

    7. +-inverses N/A

       \[\leadsto \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0}} \cdot \left(x \cdot x\right) \]

    8. metadata-eval N/A

       \[\leadsto \frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}} \cdot \left(x \cdot x\right) \]

    9. flip-- N/A

       \[\leadsto \color{blue}{\left(0 - 0\right)} \cdot \left(x \cdot x\right) \]

    10. metadata-eval N/A

        \[\leadsto \color{blue}{0} \cdot \left(x \cdot x\right) \]

    11. mul0-lft 2.6

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

11. Applied egg-rr 2.6%

    \[\leadsto \color{blue}{0} \]
12. 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 2024196 
(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)))