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

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 5

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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, 0.0× 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. add-log-exp 33.6%

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

   2. exp-prod 33.5%

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

   3. pow-sub 18.4%

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

   4. exp-prod 18.4%

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

   5. pow1 18.4%

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

   6. log1p-expm1-u 18.4%

      \[\leadsto \log \left(\frac{e^{x \cdot x}}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)}}}\right) \]

   7. log1p-undefine 18.4%

      \[\leadsto \log \left(\frac{e^{x \cdot x}}{e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(x\right)\right)}}}\right) \]

   8. add-exp-log 18.4%

      \[\leadsto \log \left(\frac{e^{x \cdot x}}{\color{blue}{1 + \mathsf{expm1}\left(x\right)}}\right) \]

   9. diff-log 18.4%

      \[\leadsto \color{blue}{\log \left(e^{x \cdot x}\right) - \log \left(1 + \mathsf{expm1}\left(x\right)\right)} \]

   10. add-log-exp 19.0%

       \[\leadsto \color{blue}{x \cdot x} - \log \left(1 + \mathsf{expm1}\left(x\right)\right) \]

   11. log1p-undefine 60.2%

       \[\leadsto x \cdot x - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x\right)\right)} \]

   12. log1p-expm1-u 100.0%

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

   13. fma-neg 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 97.7% 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 97.6%

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

   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 97.2%

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

      1. neg-mul-1 97.2%

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

   5. Simplified 97.2%

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

4. Final simplification 97.4%

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

Alternative 3: 52.5% accurate, 0.7× 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 63.6%

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

      1. neg-mul-1 63.6%

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

   5. Simplified 63.6%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 0.5%

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

      1. neg-mul-1 0.5%

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

   5. Simplified 0.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 56.4%

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

      3. sqr-neg 56.4%

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

      4. sqrt-prod 7.0%

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

      5. add-sqr-sqrt 7.0%

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

      6. /-rgt-identity 7.0%

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

   7. Applied egg-rr 7.0%

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

   8. Taylor expanded in x around 0 7.0%

      \[\leadsto \color{blue}{x} \]
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: 3.8% accurate, 5.0× 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 46.4%

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

   1. neg-mul-1 46.4%

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

5. Simplified 46.4%

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

   1. add-sqr-sqrt 25.8%

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

   2. sqrt-unprod 44.5%

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

   3. sqr-neg 44.5%

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

   4. sqrt-prod 2.7%

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

   5. add-sqr-sqrt 3.7%

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

   6. /-rgt-identity 3.7%

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

7. Applied egg-rr 3.7%

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

8. Taylor expanded in x around 0 3.7%

   \[\leadsto \color{blue}{x} \]
9. 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 2024139 
(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)))