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

Percentage Accurate: 99.9% → 100.0%

Time: 2.7s

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: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 99.9%

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

   1. associate-*r/ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \frac{x \cdot 0.5}{y} - -0.5 \]
8. Add Preprocessing

Alternative 2: 53.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{x}{0} - -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5e-310) 0.5 (if (<= y 2.4e-158) (- (/ x 0.0) -0.5) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5e-310) {
		tmp = 0.5;
	} else if (y <= 2.4e-158) {
		tmp = (x / 0.0) - -0.5;
	} 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 (y <= (-5d-310)) then
        tmp = 0.5d0
    else if (y <= 2.4d-158) then
        tmp = (x / 0.0d0) - (-0.5d0)
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5e-310) {
		tmp = 0.5;
	} else if (y <= 2.4e-158) {
		tmp = (x / 0.0) - -0.5;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5e-310:
		tmp = 0.5
	elif y <= 2.4e-158:
		tmp = (x / 0.0) - -0.5
	else:
		tmp = 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = 0.5;
	elseif (y <= 2.4e-158)
		tmp = Float64(Float64(x / 0.0) - -0.5);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5e-310)
		tmp = 0.5;
	elseif (y <= 2.4e-158)
		tmp = (x / 0.0) - -0.5;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5e-310], 0.5, If[LessEqual[y, 2.4e-158], N[(N[(x / 0.0), $MachinePrecision] - -0.5), $MachinePrecision], 0.5]]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310 or 2.40000000000000007e-158 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 59.4%

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

   if -4.999999999999985e-310 < y < 2.40000000000000007e-158

   1. Initial program 100.0%

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

   3. Simplified 99.9%

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

      1. metadata-eval 99.9%

         \[\leadsto x \cdot \frac{\color{blue}{\frac{1}{2}}}{y} - -0.5 \]

      2. associate-/r* 99.9%

         \[\leadsto x \cdot \color{blue}{\frac{1}{2 \cdot y}} - -0.5 \]

      3. count-2 99.9%

         \[\leadsto x \cdot \frac{1}{\color{blue}{y + y}} - -0.5 \]

      4. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{y + y}} - -0.5 \]

      5. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y + y} - -0.5 \]

      6. *-un-lft-identity 100.0%

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

      7. flip-+ 0.0%

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

      8. +-inverses 0.0%

         \[\leadsto \frac{x}{\frac{\color{blue}{0}}{y - y}} - -0.5 \]

      9. +-inverses 0.0%

         \[\leadsto \frac{x}{\frac{0}{\color{blue}{0}}} - -0.5 \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{0}{0}}} - -0.5 \]

   8. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{x}{0}} - -0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{x}{0} - -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 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. Step-by-step derivation

   1. frac-2neg 100.0%

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

   2. div-inv 99.7%

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

   3. flip-+ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. distribute-neg-frac 0.0%

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

   7. +-inverses 0.0%

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

   8. metadata-eval 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. +-inverses 0.0%

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

   12. flip-+ 1.0%

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

   13. flip-+ 0.0%

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

   14. +-inverses 0.0%

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

   15. +-inverses 0.0%

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

   16. +-inverses 0.0%

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

   17. +-inverses 0.0%

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

   18. clear-num 0.0%

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

   19. +-inverses 0.0%

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

   20. +-inverses 0.0%

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

4. Applied egg-rr 0.0%

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

6. Simplified 2.1%

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

7. Final simplification 2.1%

   \[\leadsto -1 \]
8. Add Preprocessing

Alternative 5: 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 100.0%

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

3. Taylor expanded in x around 0 51.6%

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

4. Final simplification 51.6%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Developer target: 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 2024020 
(FPCore (x y)
  :name "Data.Random.Distribution.T:$ccdf from random-fu-0.2.6.2"
  :precision binary64

  :herbie-target
  (+ (* 0.5 (/ x y)) 0.5)

  (/ (+ x y) (+ y y)))