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

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

Alternatives: 3

Speedup: 1.0×

• 24.1% 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, 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: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;0 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (* x x) (if (<= x 1.0) (- 0.0 x) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * x;
	} else if (x <= 1.0) {
		tmp = 0.0 - 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 = 0.0d0 - 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 = 0.0 - 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 = 0.0 - 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(0.0 - 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 = 0.0 - 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], N[(0.0 - x), $MachinePrecision], 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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{x}\right) \]

   5. Simplified 97.9%

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

   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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 99.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{0 - x} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.4%

         \[\leadsto \mathsf{neg.f64}\left(x\right) \]

   7. Applied egg-rr 99.4%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;0 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- 0.0 x))

C

double code(double x) {
	return 0.0 - x;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0 - x

Julia

function code(x)
	return Float64(0.0 - x)
end

MATLAB

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

Wolfram

code[x_] := N[(0.0 - x), $MachinePrecision]

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

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 52.9%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{x}\right) \]

5. Simplified 52.9%

   \[\leadsto \color{blue}{0 - x} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 52.9%

      \[\leadsto \mathsf{neg.f64}\left(x\right) \]

7. Applied egg-rr 52.9%

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

8. Final simplification 52.9%

   \[\leadsto 0 - x \]
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 2024145 
(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)))