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

Percentage Accurate: 94.2% → 99.9%

Time: 5.0s

Alternatives: 5

Speedup: 1.0×

• 24.5% 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: 94.2% 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: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{y}{x} \cdot y\right)}^{-1} - 3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (pow (* (/ y x) y) -1.0) 3.0))

C

double code(double x, double y) {
	return pow(((y / x) * y), -1.0) - 3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (((y / x) * y) ** (-1.0d0)) - 3.0d0
end function

Java

public static double code(double x, double y) {
	return Math.pow(((y / x) * y), -1.0) - 3.0;
}

Python

def code(x, y):
	return math.pow(((y / x) * y), -1.0) - 3.0

Julia

function code(x, y)
	return Float64((Float64(Float64(y / x) * y) ^ -1.0) - 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (((y / x) * y) ^ -1.0) - 3.0;
end

Wolfram

code[x_, y_] := N[(N[Power[N[(N[(y / x), $MachinePrecision] * y), $MachinePrecision], -1.0], $MachinePrecision] - 3.0), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y} - 3} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. rem-exp-log N/A

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

   3. lift-log.f64 N/A

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

   4. sinh-+-cosh-rev N/A

      \[\leadsto \frac{\color{blue}{\cosh \log \left(\frac{x}{y}\right) + \sinh \log \left(\frac{x}{y}\right)}}{y} - 3 \]

   5. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\cosh \log \left(\frac{x}{y}\right) \cdot \cosh \log \left(\frac{x}{y}\right) - \sinh \log \left(\frac{x}{y}\right) \cdot \sinh \log \left(\frac{x}{y}\right)}{\cosh \log \left(\frac{x}{y}\right) - \sinh \log \left(\frac{x}{y}\right)}}}{y} - 3 \]

   6. sinh-cosh N/A

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

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{2}{2}}}{\cosh \log \left(\frac{x}{y}\right) - \sinh \log \left(\frac{x}{y}\right)}}{y} - 3 \]

   8. sinh---cosh-rev N/A

      \[\leadsto \frac{\frac{\frac{2}{2}}{\color{blue}{e^{\mathsf{neg}\left(\log \left(\frac{x}{y}\right)\right)}}}}{y} - 3 \]

   9. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{e^{\mathsf{neg}\left(\log \left(\frac{x}{y}\right)\right)} \cdot y}} - 3 \]

   10. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{2}{2}}{e^{\mathsf{neg}\left(\log \left(\frac{x}{y}\right)\right)} \cdot y}} - 3 \]

   11. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{1}}{e^{\mathsf{neg}\left(\log \left(\frac{x}{y}\right)\right)} \cdot y} - 3 \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{neg}\left(\log \left(\frac{x}{y}\right)\right)} \cdot y}} - 3 \]

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{y}{x}} \cdot y} - 3 \]
8. Step-by-step derivation

   1. lower-/.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

    \[\leadsto {\left(\frac{y}{x} \cdot y\right)}^{-1} - 3 \]
11. Add Preprocessing

Alternative 2: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.8e-143) (/ (/ x y) y) (- (/ x (* y y)) 3.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.8e-143) {
		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 <= 7.8d-143) 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 <= 7.8e-143) {
		tmp = (x / y) / y;
	} else {
		tmp = (x / (y * y)) - 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.8e-143:
		tmp = (x / y) / y
	else:
		tmp = (x / (y * y)) - 3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.8e-143)
		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 <= 7.8e-143)
		tmp = (x / y) / y;
	else
		tmp = (x / (y * y)) - 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.8e-143], 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 y < 7.80000000000000007e-143

   1. Initial program 90.8%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(\frac{x}{y \cdot y}\right)}^{\color{blue}{\left(\frac{2}{2}\right)}} - 3 \]

      3. sqrt-pow1 N/A

         \[\leadsto \color{blue}{\sqrt{{\left(\frac{x}{y \cdot y}\right)}^{2}}} - 3 \]

      4. pow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. sqrt-div N/A

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

      10. rem-square-sqrt N/A

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

      11. sqrt-prod N/A

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

      12. sqr-neg-rev N/A

          \[\leadsto \frac{\sqrt{\frac{x}{y} \cdot \frac{x}{y \cdot y}}}{\sqrt{\sqrt{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)}}}} - 3 \]

      13. pow2 N/A

          \[\leadsto \frac{\sqrt{\frac{x}{y} \cdot \frac{x}{y \cdot y}}}{\sqrt{\sqrt{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{2}}}}} - 3 \]

      14. sqrt-pow1 N/A

          \[\leadsto \frac{\sqrt{\frac{x}{y} \cdot \frac{x}{y \cdot y}}}{\sqrt{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{\left(\frac{2}{2}\right)}}}} - 3 \]

      15. metadata-eval N/A

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

      16. unpow1 N/A

          \[\leadsto \frac{\sqrt{\frac{x}{y} \cdot \frac{x}{y \cdot y}}}{\sqrt{\color{blue}{\mathsf{neg}\left(y\right)}}} - 3 \]

      17. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\sqrt{\frac{x}{y} \cdot \frac{x}{y \cdot y}}}{\sqrt{\mathsf{neg}\left(y\right)}}} - 3 \]

   4. Applied rewrites 10.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{{\left(\frac{x}{y}\right)}^{2}}{y}}}{\sqrt{y}}} - 3 \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{{y}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. rem-square-sqrt N/A

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

      3. unpow2 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. unpow2 N/A

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

      11. sqr-abs-rev N/A

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

      12. associate-/r* N/A

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

      13. distribute-neg-frac N/A

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

   7. Applied rewrites 58.4%

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

   if 7.80000000000000007e-143 < 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 3: 99.9% accurate, 0.8× 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 94.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 4: 94.2% 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]

Derivation

1. Initial program 94.4%

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

Alternative 5: 51.6% accurate, 20.0× 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 94.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-3} \]
4. Step-by-step derivation

5. Applied rewrites 56.2%

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

Developer Target 1: 99.9% accurate, 0.8× 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 2024338 
(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))