Statistics.Sample:$skurtosis from math-functions-0.1.5.2

Percentage Accurate: 94.5% → 99.7%

Time: 5.3s

Alternatives: 4

Speedup: 1.0×

• 25.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot y} - 3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ x (* y y)) 3.0))

C

double code(double x, double y) {
	return (x / (y * y)) - 3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / (y * y)) - 3.0d0
end function

Java

public static double code(double x, double y) {
	return (x / (y * y)) - 3.0;
}

Python

def code(x, y):
	return (x / (y * y)) - 3.0

Julia

function code(x, y)
	return Float64(Float64(x / Float64(y * y)) - 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (x / (y * y)) - 3.0;
end

Wolfram

code[x_, y_] := N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]

Initial Program: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot y} - 3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ x (* y y)) 3.0))

C

double code(double x, double y) {
	return (x / (y * y)) - 3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / (y * y)) - 3.0d0
end function

Java

public static double code(double x, double y) {
	return (x / (y * y)) - 3.0;
}

Python

def code(x, y):
	return (x / (y * y)) - 3.0

Julia

function code(x, y)
	return Float64(Float64(x / Float64(y * y)) - 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (x / (y * y)) - 3.0;
end

Wolfram

code[x_, y_] := N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-321}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-321) (/ (/ x y) y) (- (/ x (* y y)) 3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-321) {
		tmp = (x / y) / y;
	} else {
		tmp = (x / (y * y)) - 3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d-321) then
        tmp = (x / y) / y
    else
        tmp = (x / (y * y)) - 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-321) {
		tmp = (x / y) / y;
	} else {
		tmp = (x / (y * y)) - 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-321:
		tmp = (x / y) / y
	else:
		tmp = (x / (y * y)) - 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-321)
		tmp = Float64(Float64(x / y) / y);
	else
		tmp = Float64(Float64(x / Float64(y * y)) - 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-321)
		tmp = (x / y) / y;
	else
		tmp = (x / (y * y)) - 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-321], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.99994e-321

   1. Initial program 74.9%

      \[\frac{x}{y \cdot y} - 3 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 74.9%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. *-lft-identity 74.9%

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

      2. associate-*l/ 74.4%

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

      3. unpow2 74.4%

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

      4. associate-/r* 74.4%

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

      5. *-rgt-identity 74.4%

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

      6. associate-/l* 74.4%

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

      7. unpow-1 74.4%

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

      8. unpow-1 74.4%

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

      9. pow-sqr 74.4%

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

      10. metadata-eval 74.4%

          \[\leadsto {y}^{\color{blue}{-2}} \cdot x \]

      11. *-commutative 74.4%

          \[\leadsto \color{blue}{x \cdot {y}^{-2}} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{x \cdot {y}^{-2}} \]
   6. Step-by-step derivation

      1. metadata-eval 74.4%

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

      2. pow-sqr 74.4%

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

      3. inv-pow 74.4%

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

      4. inv-pow 74.4%

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

      5. un-div-inv 74.4%

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

   7. Applied egg-rr 74.4%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{y}}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.9%

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

      2. div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   if 4.99994e-321 < (*.f64 y y)

   1. Initial program 99.9%

      \[\frac{x}{y \cdot y} - 3 \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 69.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 125000000:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;-3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 125000000.0) (/ (/ x y) y) -3.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 125000000.0) {
		tmp = (x / y) / y;
	} else {
		tmp = -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 125000000.0d0) then
        tmp = (x / y) / y
    else
        tmp = -3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 125000000.0) {
		tmp = (x / y) / y;
	} else {
		tmp = -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 125000000.0:
		tmp = (x / y) / y
	else:
		tmp = -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 125000000.0)
		tmp = Float64(Float64(x / y) / y);
	else
		tmp = -3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 125000000.0)
		tmp = (x / y) / y;
	else
		tmp = -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 125000000.0], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision], -3.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.25e8

   1. Initial program 92.4%

      \[\frac{x}{y \cdot y} - 3 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 55.0%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. *-lft-identity 55.0%

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

      2. associate-*l/ 54.8%

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

      3. unpow2 54.8%

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

      4. associate-/r* 54.8%

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

      5. *-rgt-identity 54.8%

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

      6. associate-/l* 54.7%

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

      7. unpow-1 54.7%

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

      8. unpow-1 54.7%

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

      9. pow-sqr 54.8%

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

      10. metadata-eval 54.8%

          \[\leadsto {y}^{\color{blue}{-2}} \cdot x \]

      11. *-commutative 54.8%

          \[\leadsto \color{blue}{x \cdot {y}^{-2}} \]

   5. Simplified 54.8%

      \[\leadsto \color{blue}{x \cdot {y}^{-2}} \]
   6. Step-by-step derivation

      1. metadata-eval 54.8%

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

      2. pow-sqr 54.7%

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

      3. inv-pow 54.7%

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

      4. inv-pow 54.7%

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

      5. un-div-inv 54.8%

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

   7. Applied egg-rr 54.8%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{y}}{y}} \]
   8. Step-by-step derivation

      1. associate-*r/ 62.4%

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

      2. div-inv 62.5%

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

   9. Applied egg-rr 62.5%

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

   if 1.25e8 < y

   1. Initial program 100.0%

      \[\frac{x}{y \cdot y} - 3 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 90.5%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{y}}{y} - 3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ (/ x y) y) 3.0))

C

double code(double x, double y) {
	return ((x / y) / y) - 3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / y) / y) - 3.0d0
end function

Java

public static double code(double x, double y) {
	return ((x / y) / y) - 3.0;
}

Python

def code(x, y):
	return ((x / y) / y) - 3.0

Julia

function code(x, y)
	return Float64(Float64(Float64(x / y) / y) - 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / y) / y) - 3.0;
end

Wolfram

code[x_, y_] := N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] - 3.0), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

   1. associate-/r* 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} - 3 \]

   2. clear-num 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\frac{x}{y}}}} - 3 \]

   3. inv-pow 99.9%

      \[\leadsto \color{blue}{{\left(\frac{y}{\frac{x}{y}}\right)}^{-1}} - 3 \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(\frac{y}{\frac{x}{y}}\right)}^{-1}} - 3 \]
5. Step-by-step derivation

   1. unpow-1 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\frac{x}{y}}}} - 3 \]

   2. clear-num 99.9%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} - 3 \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} - 3 \]
7. Add Preprocessing

Alternative 4: 50.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -3.0)

C

double code(double x, double y) {
	return -3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -3.0d0
end function

Java

public static double code(double x, double y) {
	return -3.0;
}

Python

def code(x, y):
	return -3.0

Julia

function code(x, y)
	return -3.0
end

MATLAB

function tmp = code(x, y)
	tmp = -3.0;
end

Wolfram

code[x_, y_] := -3.0

Derivation

1. Initial program 94.2%

   \[\frac{x}{y \cdot y} - 3 \]
2. Add Preprocessing

3. Taylor expanded in x around 0 50.8%

   \[\leadsto \color{blue}{-3} \]
4. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{y}}{y} - 3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (/ (/ x y) y) 3.0))

C

double code(double x, double y) {
	return ((x / y) / y) - 3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / y) / y) - 3.0d0
end function

Java

public static double code(double x, double y) {
	return ((x / y) / y) - 3.0;
}

Python

def code(x, y):
	return ((x / y) / y) - 3.0

Julia

function code(x, y)
	return Float64(Float64(Float64(x / y) / y) - 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / y) / y) - 3.0;
end

Wolfram

code[x_, y_] := N[(N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision] - 3.0), $MachinePrecision]

Reproduce

herbie shell --seed 2024146 
(FPCore (x y)
  :name "Statistics.Sample:$skurtosis from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (/ x y) y) 3))

  (- (/ x (* y y)) 3.0))