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

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 4

Speedup: 1.0×

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

Specification

Math

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

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

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

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 2: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.0))) (* x x) (- x)))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = x * x
    else
        tmp = -x
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.0):
		tmp = x * x
	else:
		tmp = -x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(x * x), $MachinePrecision], (-x)]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 97.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.7%

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

      1. neg-mul-1 97.7%

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

   5. Simplified 97.7%

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

4. Final simplification 97.6%

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

Alternative 3: 50.6% accurate, 2.5× 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 Float64(-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 48.6%

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

   1. neg-mul-1 48.6%

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

5. Simplified 48.6%

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

Alternative 4: 3.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0 48.6%

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

   1. neg-mul-1 48.6%

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

5. Simplified 48.6%

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

   1. add-sqr-sqrt 26.5%

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

   2. sqrt-unprod 45.9%

      \[\leadsto \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}} \]

   3. sqr-neg 45.9%

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

   4. sqrt-prod 2.7%

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

   5. add-log-exp 15.2%

      \[\leadsto \color{blue}{\log \left(e^{\sqrt{x} \cdot \sqrt{x}}\right)} \]

   6. add-sqr-sqrt 16.5%

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

   7. add-sqr-sqrt 16.5%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{x}} \cdot \sqrt{e^{x}}\right)} \]

   8. sqrt-unprod 16.5%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{x} \cdot e^{x}}\right)} \]

   9. *-un-lft-identity 16.5%

      \[\leadsto \log \left(\sqrt{e^{\color{blue}{1 \cdot x}} \cdot e^{x}}\right) \]

   10. exp-prod 16.5%

       \[\leadsto \log \left(\sqrt{\color{blue}{{\left(e^{1}\right)}^{x}} \cdot e^{x}}\right) \]

   11. pow1 16.5%

       \[\leadsto \log \left(\sqrt{{\left(e^{1}\right)}^{\color{blue}{\left({x}^{1}\right)}} \cdot e^{x}}\right) \]

   12. metadata-eval 16.5%

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

   13. sqrt-pow1 16.5%

       \[\leadsto \log \left(\sqrt{{\left(e^{1}\right)}^{\color{blue}{\left(\sqrt{{x}^{2}}\right)}} \cdot e^{x}}\right) \]

   14. pow2 16.5%

       \[\leadsto \log \left(\sqrt{{\left(e^{1}\right)}^{\left(\sqrt{\color{blue}{x \cdot x}}\right)} \cdot e^{x}}\right) \]

   15. sqr-neg 16.5%

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

   16. sqrt-unprod 1.4%

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

   17. add-sqr-sqrt 2.6%

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

   18. exp-prod 2.6%

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

   19. *-un-lft-identity 2.6%

       \[\leadsto \log \left(\sqrt{e^{\color{blue}{-x}} \cdot e^{x}}\right) \]

   20. exp-neg 2.6%

       \[\leadsto \log \left(\sqrt{\color{blue}{\frac{1}{e^{x}}} \cdot e^{x}}\right) \]

   21. lft-mult-inverse 3.5%

       \[\leadsto \log \left(\sqrt{\color{blue}{1}}\right) \]

   22. metadata-eval 3.5%

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

7. Applied egg-rr 3.5%

   \[\leadsto \color{blue}{0} \]
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 2024191 
(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)))