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

Percentage Accurate: 100.0% → 100.0%

Time: 12.5s

Alternatives: 3

Speedup: 1.0×

• 24.6% 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, 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 35.2%

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

   2. exp-prod 35.2%

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

   3. pow-sub 20.5%

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

   4. exp-prod 20.5%

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

   5. pow1 20.5%

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

   6. log1p-expm1-u 20.5%

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

   7. log1p-undefine 20.5%

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

   8. add-exp-log 20.5%

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

   9. diff-log 20.5%

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

   10. add-log-exp 20.6%

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

   11. log1p-undefine 61.7%

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

   12. log1p-expm1-u 100.0%

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

   13. fmm-def 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: 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 3: 3.7% 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. Step-by-step derivation

   1. add-log-exp 35.2%

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

   2. exp-prod 35.2%

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

   3. pow-sub 20.5%

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

   4. exp-prod 20.5%

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

   5. pow1 20.5%

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

   6. log1p-expm1-u 20.5%

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

   7. log1p-undefine 20.5%

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

   8. add-exp-log 20.5%

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

   9. diff-log 20.5%

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

   10. add-log-exp 20.6%

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

   11. log1p-undefine 61.7%

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

   12. log1p-expm1-u 100.0%

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

   13. fmm-def 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 3.5%

   \[\leadsto \color{blue}{0} \]
7. 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 2024179 
(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)))