Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 4.3s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ x \cdot \log x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (log x)))

C

double code(double x) {
	return x * log(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * log(x)
end function

Java

public static double code(double x) {
	return x * Math.log(x);
}

Python

def code(x):
	return x * math.log(x)

Julia

function code(x)
	return Float64(x * log(x))
end

MATLAB

function tmp = code(x)
	tmp = x * log(x);
end

Wolfram

code[x_] := N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \log x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (log x)))

C

double code(double x) {
	return x * log(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * log(x)
end function

Java

public static double code(double x) {
	return x * Math.log(x);
}

Python

def code(x):
	return x * math.log(x)

Julia

function code(x)
	return Float64(x * log(x))
end

MATLAB

function tmp = code(x)
	tmp = x * log(x);
end

Wolfram

code[x_] := N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (- (log (/ 1.0 x)))))

C

double code(double x) {
	return x * -log((1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * -log((1.0d0 / x))
end function

Java

public static double code(double x) {
	return x * -Math.log((1.0 / x));
}

Python

def code(x):
	return x * -math.log((1.0 / x))

Julia

function code(x)
	return Float64(x * Float64(-log(Float64(1.0 / x))))
end

MATLAB

function tmp = code(x)
	tmp = x * -log((1.0 / x));
end

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x \cdot \log x \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.6%

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

4. Final simplification 99.6%

   \[\leadsto x \cdot \left(-\log \left(\frac{1}{x}\right)\right) \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \log x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (log x)))

C

double code(double x) {
	return x * log(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * log(x)
end function

Java

public static double code(double x) {
	return x * Math.log(x);
}

Python

def code(x):
	return x * math.log(x)

Julia

function code(x)
	return Float64(x * log(x))
end

MATLAB

function tmp = code(x)
	tmp = x * log(x);
end

Wolfram

code[x_] := N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x \cdot \log x \]
2. Add Preprocessing

3. Final simplification 99.6%

   \[\leadsto x \cdot \log x \]
4. Add Preprocessing

Alternative 3: 3.7% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[x \cdot \log x \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.6%

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \log \left(\frac{1}{x}\right)\right)} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 99.6%

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

   2. associate-/r* 99.6%

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

   3. metadata-eval 99.6%

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

   4. sqrt-div 99.6%

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

   5. add-exp-log 99.6%

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

   6. add-sqr-sqrt 48.2%

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

   7. sqrt-unprod 50.9%

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

   8. log-rec 50.9%

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

   9. log-rec 50.9%

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

   10. sqr-neg 50.9%

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

   11. sqrt-unprod 2.8%

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

   12. add-sqr-sqrt 6.0%

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

   13. add-exp-log 3.6%

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

   14. /-rgt-identity 3.6%

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

   15. /-rgt-identity 3.6%

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

   16. log-div 3.6%

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

5. Applied egg-rr 3.6%

   \[\leadsto -1 \cdot \left(x \cdot \color{blue}{\left(\log \left(\sqrt{x}\right) - \log \left(\sqrt{x}\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-inverses 3.6%

      \[\leadsto -1 \cdot \left(x \cdot \color{blue}{0}\right) \]

7. Simplified 3.6%

   \[\leadsto -1 \cdot \left(x \cdot \color{blue}{0}\right) \]

8. Final simplification 3.6%

   \[\leadsto x \cdot 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))