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

Percentage Accurate: 99.9% → 100.0%

Time: 3.2s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. --rgt-identity 100.0%

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

   3. associate-+l- 100.0%

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

   4. neg-sub0 100.0%

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

   5. div-sub 100.0%

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

   6. sub-neg 100.0%

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

   7. distribute-frac-neg 100.0%

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

   8. neg-mul-1 100.0%

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

   9. associate-*l/ 99.9%

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

   10. *-commutative 99.9%

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

   11. cancel-sign-sub 99.9%

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

   12. count-2 99.9%

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

   13. *-commutative 99.9%

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

   14. associate-/r* 99.9%

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

   15. *-inverses 99.9%

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

   16. metadata-eval 99.9%

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

   17. remove-double-neg 99.9%

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

   18. count-2 99.9%

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

   19. associate-/r* 99.9%

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

   20. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. associate-*r/ 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -360:\\ \;\;\;\;0.5\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -360.0) 0.5 (if (<= y 3.1e+26) (* x (/ 0.5 y)) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -360.0) {
		tmp = 0.5;
	} else if (y <= 3.1e+26) {
		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 (y <= (-360.0d0)) then
        tmp = 0.5d0
    else if (y <= 3.1d+26) 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 (y <= -360.0) {
		tmp = 0.5;
	} else if (y <= 3.1e+26) {
		tmp = x * (0.5 / y);
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -360.0:
		tmp = 0.5
	elif y <= 3.1e+26:
		tmp = x * (0.5 / y)
	else:
		tmp = 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -360.0)
		tmp = 0.5;
	elseif (y <= 3.1e+26)
		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 (y <= -360.0)
		tmp = 0.5;
	elseif (y <= 3.1e+26)
		tmp = x * (0.5 / y);
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -360.0], 0.5, If[LessEqual[y, 3.1e+26], N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], 0.5]]

Derivation

1. Split input into 2 regimes

2. if y < -360 or 3.1e26 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. --rgt-identity 100.0%

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

      3. associate-+l- 100.0%

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

      4. neg-sub0 100.0%

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

      5. div-sub 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*l/ 99.9%

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

      10. *-commutative 99.9%

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

      11. cancel-sign-sub 99.9%

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

      12. count-2 99.9%

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

      13. *-commutative 99.9%

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

      14. associate-/r* 99.9%

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

      15. *-inverses 99.9%

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

      16. metadata-eval 99.9%

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

      17. remove-double-neg 99.9%

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

      18. count-2 99.9%

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

      19. associate-/r* 99.9%

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

      20. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 83.1%

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

   if -360 < y < 3.1e26

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. --rgt-identity 100.0%

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

      3. associate-+l- 100.0%

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

      4. neg-sub0 100.0%

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

      5. div-sub 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*l/ 99.8%

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

      10. *-commutative 99.8%

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

      11. cancel-sign-sub 99.8%

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

      12. count-2 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.8%

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

      15. *-inverses 99.8%

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

      16. metadata-eval 99.8%

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

      17. remove-double-neg 99.8%

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

      18. count-2 99.8%

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

      19. associate-/r* 99.8%

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

      20. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. associate-*r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 74.8%

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

      1. associate-*r/ 74.8%

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

      2. *-rgt-identity 74.8%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot x\right) \cdot 1}}{y} \]

      3. associate-*r/ 74.6%

         \[\leadsto \color{blue}{\left(0.5 \cdot x\right) \cdot \frac{1}{y}} \]

      4. *-commutative 74.6%

         \[\leadsto \color{blue}{\left(x \cdot 0.5\right)} \cdot \frac{1}{y} \]

      5. associate-*l* 74.6%

         \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \frac{1}{y}\right)} \]

      6. associate-*r/ 74.6%

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

      7. metadata-eval 74.6%

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

   8. Simplified 74.6%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -360:\\ \;\;\;\;0.5\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+26}:\\ \;\;\;\;x \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 3: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13.5:\\ \;\;\;\;0.5\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -13.5) 0.5 (if (<= y 2.2e+27) (/ (* 0.5 x) y) 0.5)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -13.5) {
		tmp = 0.5;
	} else if (y <= 2.2e+27) {
		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) :: tmp
    if (y <= (-13.5d0)) then
        tmp = 0.5d0
    else if (y <= 2.2d+27) 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 tmp;
	if (y <= -13.5) {
		tmp = 0.5;
	} else if (y <= 2.2e+27) {
		tmp = (0.5 * x) / y;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -13.5:
		tmp = 0.5
	elif y <= 2.2e+27:
		tmp = (0.5 * x) / y
	else:
		tmp = 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -13.5)
		tmp = 0.5;
	elseif (y <= 2.2e+27)
		tmp = Float64(Float64(0.5 * x) / y);
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -13.5)
		tmp = 0.5;
	elseif (y <= 2.2e+27)
		tmp = (0.5 * x) / y;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -13.5], 0.5, If[LessEqual[y, 2.2e+27], N[(N[(0.5 * x), $MachinePrecision] / y), $MachinePrecision], 0.5]]

Derivation

1. Split input into 2 regimes

2. if y < -13.5 or 2.1999999999999999e27 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. --rgt-identity 100.0%

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

      3. associate-+l- 100.0%

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

      4. neg-sub0 100.0%

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

      5. div-sub 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*l/ 99.9%

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

      10. *-commutative 99.9%

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

      11. cancel-sign-sub 99.9%

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

      12. count-2 99.9%

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

      13. *-commutative 99.9%

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

      14. associate-/r* 99.9%

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

      15. *-inverses 99.9%

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

      16. metadata-eval 99.9%

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

      17. remove-double-neg 99.9%

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

      18. count-2 99.9%

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

      19. associate-/r* 99.9%

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

      20. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 83.1%

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

   if -13.5 < y < 2.1999999999999999e27

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. --rgt-identity 100.0%

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

      3. associate-+l- 100.0%

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

      4. neg-sub0 100.0%

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

      5. div-sub 100.0%

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

      6. sub-neg 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. neg-mul-1 100.0%

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

      9. associate-*l/ 99.8%

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

      10. *-commutative 99.8%

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

      11. cancel-sign-sub 99.8%

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

      12. count-2 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.8%

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

      15. *-inverses 99.8%

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

      16. metadata-eval 99.8%

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

      17. remove-double-neg 99.8%

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

      18. count-2 99.8%

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

      19. associate-/r* 99.8%

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

      20. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 74.8%

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

   6. Simplified 74.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13.5:\\ \;\;\;\;0.5\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{0.5 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. +-commutative 100.0%

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

   2. --rgt-identity 100.0%

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

   3. associate-+l- 100.0%

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

   4. neg-sub0 100.0%

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

   5. div-sub 100.0%

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

   6. sub-neg 100.0%

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

   7. distribute-frac-neg 100.0%

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

   8. neg-mul-1 100.0%

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

   9. associate-*l/ 99.9%

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

   10. *-commutative 99.9%

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

   11. cancel-sign-sub 99.9%

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

   12. count-2 99.9%

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

   13. *-commutative 99.9%

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

   14. associate-/r* 99.9%

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

   15. *-inverses 99.9%

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

   16. metadata-eval 99.9%

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

   17. remove-double-neg 99.9%

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

   18. count-2 99.9%

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

   19. associate-/r* 99.9%

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

   20. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 5: 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. 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%

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

   5. +-inverses 0.0%

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

   6. distribute-neg-frac 0.0%

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

   7. +-inverses 0.0%

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

   8. metadata-eval 0.0%

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

   9. +-inverses 0.0%

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

   10. flip-+ 1.1%

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

   11. flip-+ 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. +-inverses 0.0%

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

   15. +-inverses 0.0%

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

   16. clear-num 0.0%

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

   17. +-inverses 0.0%

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

   18. +-inverses 0.0%

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

3. Applied egg-rr 0.0%

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

5. Simplified 2.1%

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

6. Final simplification 2.1%

   \[\leadsto -1 \]

Alternative 6: 50.7% 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. Step-by-step derivation

   1. +-commutative 100.0%

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

   2. --rgt-identity 100.0%

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

   3. associate-+l- 100.0%

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

   4. neg-sub0 100.0%

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

   5. div-sub 100.0%

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

   6. sub-neg 100.0%

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

   7. distribute-frac-neg 100.0%

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

   8. neg-mul-1 100.0%

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

   9. associate-*l/ 99.9%

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

   10. *-commutative 99.9%

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

   11. cancel-sign-sub 99.9%

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

   12. count-2 99.9%

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

   13. *-commutative 99.9%

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

   14. associate-/r* 99.9%

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

   15. *-inverses 99.9%

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

   16. metadata-eval 99.9%

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

   17. remove-double-neg 99.9%

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

   18. count-2 99.9%

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

   19. associate-/r* 99.9%

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

   20. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 53.0%

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

5. Final simplification 53.0%

   \[\leadsto 0.5 \]

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 2023336 
(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)))