Statistics.Correlation.Kendall:numOfTiesBy from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 3.6s

Alternatives: 5

Speedup: 1.0×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (- x 1.0)))

C

double code(double x) {
	return x * (x - 1.0);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x - 1.0);
}

Python

def code(x):
	return x * (x - 1.0)

Julia

function code(x)
	return Float64(x * Float64(x - 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x - 1.0);
end

Wolfram

code[x_] := N[(x * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (- x 1.0)))

C

double code(double x) {
	return x * (x - 1.0);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x - 1.0);
}

Python

def code(x):
	return x * (x - 1.0)

Julia

function code(x)
	return Float64(x * Float64(x - 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x - 1.0);
end

Wolfram

code[x_] := N[(x * N[(x - 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, -x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x x (- x)))

C

double code(double x) {
	return fma(x, x, -x);
}

Julia

function code(x)
	return fma(x, x, Float64(-x))
end

Wolfram

code[x_] := N[(x * x + (-x)), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot \left(x - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

   3. lower-fma.f64 N/A

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

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{-1} \cdot x\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{neg}\left(x\right)}\right) \]

   6. lower-neg.f64 100.0

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{-x}\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -x\right)} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \left(x + -1\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x (+ x -1.0)) 4e-6) (- x) (* x x)))

C

double code(double x) {
	double tmp;
	if ((x * (x + -1.0)) <= 4e-6) {
		tmp = -x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * (x + (-1.0d0))) <= 4d-6) then
        tmp = -x
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * (x + -1.0)) <= 4e-6) {
		tmp = -x;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * (x + -1.0)) <= 4e-6:
		tmp = -x
	else:
		tmp = x * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * Float64(x + -1.0)) <= 4e-6)
		tmp = Float64(-x);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * (x + -1.0)) <= 4e-6)
		tmp = -x;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], 4e-6], (-x), N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 x #s(literal 1 binary64))) < 3.99999999999999982e-6

   1. Initial program 100.0%

      \[x \cdot \left(x - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 97.2

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

   5. Applied rewrites 97.2%

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

   if 3.99999999999999982e-6 < (*.f64 x (-.f64 x #s(literal 1 binary64)))

   1. Initial program 100.0%

      \[x \cdot \left(x - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(x + -1\right) \leq 4 \cdot 10^{-6}:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) (- x) x))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = -x
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = -x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = -x
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(-x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = -x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], (-x), x]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

      \[x \cdot \left(x - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if 1 < x

   1. Initial program 100.0%

      \[x \cdot \left(x - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 0.5

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

   5. Applied rewrites 0.5%

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

      1. neg-sub0 N/A

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

      2. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

      4. sqr-pow N/A

         \[\leadsto \frac{0 - \color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

      5. unpow-prod-down N/A

         \[\leadsto \frac{0 - \color{blue}{{\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

      6. sqr-neg N/A

         \[\leadsto \frac{0 - {\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{0 - {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{0 - {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

      9. unpow-prod-down N/A

         \[\leadsto \frac{0 - \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

      10. sqr-pow N/A

          \[\leadsto \frac{0 - \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

      11. unpow3 N/A

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

      12. lift-neg.f64 N/A

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

      13. lift-neg.f64 N/A

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

      14. sqr-neg N/A

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

      15. lift-*.f64 N/A

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

   7. Applied rewrites 6.8%

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

Alternative 4: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x + -1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ x -1.0)))

C

double code(double x) {
	return x * (x + -1.0);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x + -1.0);
}

Python

def code(x):
	return x * (x + -1.0)

Julia

function code(x)
	return Float64(x * Float64(x + -1.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x + -1.0);
end

Wolfram

code[x_] := N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot \left(x - 1\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x \cdot \left(x + -1\right) \]
4. Add Preprocessing

Alternative 5: 3.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 100.0%

   \[x \cdot \left(x - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 52.3

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

5. Applied rewrites 52.3%

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

   1. neg-sub0 N/A

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

   2. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   4. sqr-pow N/A

      \[\leadsto \frac{0 - \color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{0 - \color{blue}{{\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   6. sqr-neg N/A

      \[\leadsto \frac{0 - {\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   7. lift-neg.f64 N/A

      \[\leadsto \frac{0 - {\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   8. lift-neg.f64 N/A

      \[\leadsto \frac{0 - {\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   9. unpow-prod-down N/A

      \[\leadsto \frac{0 - \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   10. sqr-pow N/A

       \[\leadsto \frac{0 - \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)} \]

   11. unpow3 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-neg.f64 N/A

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

   14. sqr-neg N/A

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

   15. lift-*.f64 N/A

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

7. Applied rewrites 3.9%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- (* x x) x))

C

double code(double x) {
	return (x * x) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) - x
end function

Java

public static double code(double x) {
	return (x * x) - x;
}

Python

def code(x):
	return (x * x) - x

Julia

function code(x)
	return Float64(Float64(x * x) - x)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) - x;
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "Statistics.Correlation.Kendall:numOfTiesBy from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x x) x))

  (* x (- x 1.0)))