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

Percentage Accurate: 100.0% → 100.0%

Time: 1.5s

Alternatives: 4

Speedup: 1.0×

• 25.1% 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. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{-1 \cdot x + {x}^{2}} \]
3. Step-by-step derivation

   1. neg-mul-1 100.0%

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

   2. unpow2 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-def 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 97.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. Taylor expanded in x around inf 96.4%

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

      1. unpow2 96.4%

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

   4. Simplified 96.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[x \cdot \left(x - 1\right) \]

   2. Taylor expanded in x around 0 97.2%

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

      1. neg-mul-1 97.2%

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

   4. Simplified 97.2%

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

4. Final simplification 96.8%

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

Alternative 3: 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. Final simplification 100.0%

   \[\leadsto x \cdot \left(x + -1\right) \]

Alternative 4: 51.1% 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. Taylor expanded in x around 0 52.0%

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

   1. neg-mul-1 52.0%

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

4. Simplified 52.0%

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

5. Final simplification 52.0%

   \[\leadsto -x \]

Developer target: 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 2023229 
(FPCore (x)
  :name "Statistics.Correlation.Kendall:numOfTiesBy from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (* x x) x)

  (* x (- x 1.0)))