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

Percentage Accurate: 99.6% → 99.6%

Time: 8.9s

Alternatives: 4

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot 0.3333333333333333, \log x, \log \left(\sqrt[3]{x}\right) \cdot \left(x \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* x 0.3333333333333333) (log x) (* (log (cbrt x)) (* x 2.0))))

C

double code(double x) {
	return fma((x * 0.3333333333333333), log(x), (log(cbrt(x)) * (x * 2.0)));
}

Julia

function code(x)
	return fma(Float64(x * 0.3333333333333333), log(x), Float64(log(cbrt(x)) * Float64(x * 2.0)))
end

Wolfram

code[x_] := N[(N[(x * 0.3333333333333333), $MachinePrecision] * N[Log[x], $MachinePrecision] + N[(N[Log[N[Power[x, 1/3], $MachinePrecision]], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x \cdot \log x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 99.6%

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

   2. pow2 99.6%

      \[\leadsto x \cdot \log \color{blue}{\left({\left(\sqrt{x}\right)}^{2}\right)} \]

   3. log-pow 99.6%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{x}\right)\right)} \]

4. Applied egg-rr 99.6%

   \[\leadsto x \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{x}\right)\right)} \]
5. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto x \cdot \left(2 \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right) \cdot \sqrt[3]{\sqrt{x}}\right)}\right) \]

   2. pow3 99.6%

      \[\leadsto x \cdot \left(2 \cdot \log \color{blue}{\left({\left(\sqrt[3]{\sqrt{x}}\right)}^{3}\right)}\right) \]

   3. log-pow 99.4%

      \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(3 \cdot \log \left(\sqrt[3]{\sqrt{x}}\right)\right)}\right) \]

   4. pow1 99.4%

      \[\leadsto x \cdot \left(2 \cdot \left(3 \cdot \log \color{blue}{\left({\left(\sqrt[3]{\sqrt{x}}\right)}^{1}\right)}\right)\right) \]

   5. metadata-eval 99.4%

      \[\leadsto x \cdot \left(2 \cdot \left(3 \cdot \log \left({\left(\sqrt[3]{\sqrt{x}}\right)}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right)\right)\right) \]

   6. pow-pow 99.4%

      \[\leadsto x \cdot \left(2 \cdot \left(3 \cdot \log \color{blue}{\left({\left({\left(\sqrt[3]{\sqrt{x}}\right)}^{2}\right)}^{0.5}\right)}\right)\right) \]

   7. pow2 99.4%

      \[\leadsto x \cdot \left(2 \cdot \left(3 \cdot \log \left({\color{blue}{\left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right)}}^{0.5}\right)\right)\right) \]

   8. cbrt-prod 99.4%

      \[\leadsto x \cdot \left(2 \cdot \left(3 \cdot \log \left({\color{blue}{\left(\sqrt[3]{\sqrt{x} \cdot \sqrt{x}}\right)}}^{0.5}\right)\right)\right) \]

   9. add-sqr-sqrt 99.4%

      \[\leadsto x \cdot \left(2 \cdot \left(3 \cdot \log \left({\left(\sqrt[3]{\color{blue}{x}}\right)}^{0.5}\right)\right)\right) \]

   10. pow1/3 99.2%

       \[\leadsto x \cdot \left(2 \cdot \left(3 \cdot \log \left({\color{blue}{\left({x}^{0.3333333333333333}\right)}}^{0.5}\right)\right)\right) \]

   11. pow-pow 99.2%

       \[\leadsto x \cdot \left(2 \cdot \left(3 \cdot \log \color{blue}{\left({x}^{\left(0.3333333333333333 \cdot 0.5\right)}\right)}\right)\right) \]

   12. metadata-eval 99.2%

       \[\leadsto x \cdot \left(2 \cdot \left(3 \cdot \log \left({x}^{\color{blue}{0.16666666666666666}}\right)\right)\right) \]

6. Applied egg-rr 99.2%

   \[\leadsto x \cdot \left(2 \cdot \color{blue}{\left(3 \cdot \log \left({x}^{0.16666666666666666}\right)\right)}\right) \]

7. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{6 \cdot \left(x \cdot \log \left({\left(1 \cdot x\right)}^{0.16666666666666666}\right)\right)} \]
8. Step-by-step derivation

   1. associate-*r* 99.3%

      \[\leadsto \color{blue}{\left(6 \cdot x\right) \cdot \log \left({\left(1 \cdot x\right)}^{0.16666666666666666}\right)} \]

   2. *-commutative 99.3%

      \[\leadsto \color{blue}{\left(x \cdot 6\right)} \cdot \log \left({\left(1 \cdot x\right)}^{0.16666666666666666}\right) \]

   3. *-lft-identity 99.3%

      \[\leadsto \left(x \cdot 6\right) \cdot \log \left({\color{blue}{x}}^{0.16666666666666666}\right) \]

   4. exp-to-pow 99.2%

      \[\leadsto \left(x \cdot 6\right) \cdot \log \color{blue}{\left(e^{\log x \cdot 0.16666666666666666}\right)} \]

   5. *-commutative 99.2%

      \[\leadsto \left(x \cdot 6\right) \cdot \log \left(e^{\color{blue}{0.16666666666666666 \cdot \log x}}\right) \]

   6. rem-log-exp 99.2%

      \[\leadsto \left(x \cdot 6\right) \cdot \color{blue}{\left(0.16666666666666666 \cdot \log x\right)} \]

   7. associate-*l* 99.2%

      \[\leadsto \color{blue}{x \cdot \left(6 \cdot \left(0.16666666666666666 \cdot \log x\right)\right)} \]

9. Simplified 99.2%

   \[\leadsto \color{blue}{x \cdot \left(6 \cdot \left(0.16666666666666666 \cdot \log x\right)\right)} \]
10. Step-by-step derivation

    1. associate-*r* 99.6%

       \[\leadsto x \cdot \color{blue}{\left(\left(6 \cdot 0.16666666666666666\right) \cdot \log x\right)} \]

    2. metadata-eval 99.6%

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

    3. *-un-lft-identity 99.6%

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

    4. add-cube-cbrt 99.6%

       \[\leadsto x \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} \]

    5. pow2 99.6%

       \[\leadsto x \cdot \log \left(\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot \sqrt[3]{x}\right) \]

    6. exp-to-pow 99.6%

       \[\leadsto x \cdot \log \left(\color{blue}{e^{\log \left(\sqrt[3]{x}\right) \cdot 2}} \cdot \sqrt[3]{x}\right) \]

    7. *-commutative 99.6%

       \[\leadsto x \cdot \log \left(e^{\color{blue}{2 \cdot \log \left(\sqrt[3]{x}\right)}} \cdot \sqrt[3]{x}\right) \]

    8. pow1/3 99.5%

       \[\leadsto x \cdot \log \left(e^{2 \cdot \log \left(\sqrt[3]{x}\right)} \cdot \color{blue}{{x}^{0.3333333333333333}}\right) \]

    9. pow-to-exp 99.5%

       \[\leadsto x \cdot \log \left(e^{2 \cdot \log \left(\sqrt[3]{x}\right)} \cdot \color{blue}{e^{\log x \cdot 0.3333333333333333}}\right) \]

    10. *-commutative 99.5%

        \[\leadsto x \cdot \log \left(e^{2 \cdot \log \left(\sqrt[3]{x}\right)} \cdot e^{\color{blue}{0.3333333333333333 \cdot \log x}}\right) \]

    11. sum-log 99.5%

        \[\leadsto x \cdot \color{blue}{\left(\log \left(e^{2 \cdot \log \left(\sqrt[3]{x}\right)}\right) + \log \left(e^{0.3333333333333333 \cdot \log x}\right)\right)} \]

    12. add-log-exp 99.5%

        \[\leadsto x \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{x}\right)} + \log \left(e^{0.3333333333333333 \cdot \log x}\right)\right) \]

    13. add-log-exp 99.5%

        \[\leadsto x \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right) + \color{blue}{0.3333333333333333 \cdot \log x}\right) \]

    14. +-commutative 99.5%

        \[\leadsto x \cdot \color{blue}{\left(0.3333333333333333 \cdot \log x + 2 \cdot \log \left(\sqrt[3]{x}\right)\right)} \]

    15. distribute-rgt-out 99.6%

        \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \log x\right) \cdot x + \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot x} \]

    16. *-commutative 99.6%

        \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot \log x\right)} + \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot x \]

    17. associate-*r* 99.6%

        \[\leadsto \color{blue}{\left(x \cdot 0.3333333333333333\right) \cdot \log x} + \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot x \]

    18. fma-define 99.6%

        \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 0.3333333333333333, \log x, \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot x\right)} \]

11. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 0.3333333333333333, \log x, \log \left(\sqrt[3]{x}\right) \cdot \left(2 \cdot x\right)\right)} \]

12. Final simplification 99.6%

    \[\leadsto \mathsf{fma}\left(x \cdot 0.3333333333333333, \log x, \log \left(\sqrt[3]{x}\right) \cdot \left(x \cdot 2\right)\right) \]
13. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(2 \cdot \log \left(\sqrt{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* 2.0 (log (sqrt x)))))

C

double code(double x) {
	return x * (2.0 * log(sqrt(x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (2.0d0 * log(sqrt(x)))
end function

Java

public static double code(double x) {
	return x * (2.0 * Math.log(Math.sqrt(x)));
}

Python

def code(x):
	return x * (2.0 * math.log(math.sqrt(x)))

Julia

function code(x)
	return Float64(x * Float64(2.0 * log(sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = x * (2.0 * log(sqrt(x)));
end

Wolfram

code[x_] := N[(x * N[(2.0 * N[Log[N[Sqrt[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[x \cdot \log x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 99.6%

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

   2. pow2 99.6%

      \[\leadsto x \cdot \log \color{blue}{\left({\left(\sqrt{x}\right)}^{2}\right)} \]

   3. log-pow 99.6%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{x}\right)\right)} \]

4. Applied egg-rr 99.6%

   \[\leadsto x \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{x}\right)\right)} \]
5. Add Preprocessing

Alternative 3: 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. Add Preprocessing

Alternative 4: 3.8% accurate, 103.0× 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 99.6%

   \[x \cdot \log x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 99.6%

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

   2. pow2 99.6%

      \[\leadsto x \cdot \log \color{blue}{\left({\left(\sqrt{x}\right)}^{2}\right)} \]

   3. log-pow 99.6%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{x}\right)\right)} \]

4. Applied egg-rr 99.6%

   \[\leadsto x \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{x}\right)\right)} \]
5. Step-by-step derivation

   1. log-pow 99.6%

      \[\leadsto x \cdot \color{blue}{\log \left({\left(\sqrt{x}\right)}^{2}\right)} \]

   2. pow2 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. add-sqr-sqrt 47.6%

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

   5. sqrt-unprod 49.7%

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

   6. sqr-neg 49.7%

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

   7. log-rec 49.7%

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

   8. log-rec 49.7%

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

   9. sqrt-unprod 1.9%

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

   10. add-sqr-sqrt 2.2%

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

   11. add-sqr-sqrt 2.2%

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

   12. associate-/r* 2.2%

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

6. Applied egg-rr 3.8%

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

   1. +-inverses 3.8%

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

8. Simplified 3.8%

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

9. Taylor expanded in x around 0 3.8%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (x)
  :name "Statistics.Distribution.Binomial:directEntropy from math-functions-0.1.5.2"
  :precision binary64
  (* x (log x)))