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

Percentage Accurate: 94.4% → 99.9%

Time: 6.5s

Alternatives: 4

Speedup: 1.0×

• 25.2% 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: 94.4% 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, 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 93.2%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

   1. un-div-inv 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 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{-260}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e-260) (/ (/ x y) y) (- (/ x (* y y)) 3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e-260) {
		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) <= 1d-260) 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) <= 1e-260) {
		tmp = (x / y) / y;
	} else {
		tmp = (x / (y * y)) - 3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1e-260:
		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) <= 1e-260)
		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) <= 1e-260)
		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], 1e-260], 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) < 9.99999999999999961e-261

   1. Initial program 77.5%

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

   3. Taylor expanded in x around inf 77.6%

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

      1. *-lft-identity 77.6%

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

      2. associate-*l/ 77.1%

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

      3. unpow2 77.1%

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

      4. associate-/r* 77.1%

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

      5. *-rgt-identity 77.1%

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

      6. associate-/l* 77.1%

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

      7. unpow-1 77.1%

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

      8. unpow-1 77.1%

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

      9. pow-sqr 77.1%

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

      10. metadata-eval 77.1%

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

      11. *-commutative 77.1%

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

   5. Simplified 77.1%

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

      1. metadata-eval 77.1%

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

      2. pow-flip 77.1%

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

      3. pow2 77.1%

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

      4. associate-/r* 77.1%

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

   7. Applied egg-rr 77.1%

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

      1. associate-/l/ 77.1%

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

      2. div-inv 77.5%

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

      3. associate-/l/ 99.8%

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

   9. Applied egg-rr 99.8%

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

   if 9.99999999999999961e-261 < (*.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 3: 69.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.06 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \mathbf{else}:\\ \;\;\;\;-3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.06e-19) (/ (/ x y) y) -3.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.06e-19) {
		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 <= 1.06d-19) 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 <= 1.06e-19) {
		tmp = (x / y) / y;
	} else {
		tmp = -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.06e-19:
		tmp = (x / y) / y
	else:
		tmp = -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.06e-19)
		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 <= 1.06e-19)
		tmp = (x / y) / y;
	else
		tmp = -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.06e-19], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision], -3.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.06e-19

   1. Initial program 90.4%

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

   3. Taylor expanded in x around inf 58.0%

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

      1. *-lft-identity 58.0%

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

      2. associate-*l/ 57.8%

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

      3. unpow2 57.8%

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

      4. associate-/r* 57.8%

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

      5. *-rgt-identity 57.8%

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

      6. associate-/l* 57.7%

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

      7. unpow-1 57.7%

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

      8. unpow-1 57.7%

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

      9. pow-sqr 57.9%

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

      10. metadata-eval 57.9%

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

      11. *-commutative 57.9%

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

   5. Simplified 57.9%

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

      1. metadata-eval 57.9%

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

      2. pow-flip 57.8%

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

      3. pow2 57.8%

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

      4. associate-/r* 57.8%

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

   7. Applied egg-rr 57.8%

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

      1. associate-/l/ 57.8%

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

      2. div-inv 58.0%

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

      3. associate-/l/ 67.5%

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

   9. Applied egg-rr 67.5%

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

   if 1.06e-19 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 88.2%

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

Alternative 4: 49.5% 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 93.2%

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

3. Taylor expanded in x around 0 49.8%

   \[\leadsto \color{blue}{-3} \]
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 2024131 
(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))