Jmat.Real.lambertw, estimator

Percentage Accurate: 99.6% → 100.0%

Time: 8.1s

Alternatives: 3

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-log.f64 N/A

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

   2. lift-log.f64 99.6

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

   3. remove-double-neg N/A

      \[\leadsto \log x - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \log x\right)\right)\right)\right)} \]

   4. neg-sub0 N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\color{blue}{\left(0 - \log \log x\right)}\right)\right) \]

   5. flip3-- N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {\log \log x}^{3}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{0} - {\log \log x}^{3}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   7. neg-sub0 N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({\log \log x}^{3}\right)}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   8. cube-neg N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(\log \log x\right)\right)}^{3}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   9. sqr-pow N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(\log \log x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(\log \log x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   10. unpow-prod-down N/A

       \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(\log \log x\right)\right) \cdot \left(\mathsf{neg}\left(\log \log x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   11. sqr-neg N/A

       \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(\log \log x \cdot \log \log x\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   12. pow-prod-down N/A

       \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\log \log x}^{\left(\frac{3}{2}\right)} \cdot {\log \log x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   13. sqr-pow N/A

       \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\log \log x}^{3}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   14. distribute-frac-neg N/A

       \[\leadsto \log x - \color{blue}{\frac{\mathsf{neg}\left({\log \log x}^{3}\right)}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}} \]

4. Applied rewrites 99.6%

   \[\leadsto \log x - \color{blue}{{\log \log x}^{2} \cdot \frac{1}{\log \log x}} \]
5. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \log x - {\log \color{blue}{\log x}}^{2} \cdot \frac{1}{\log \log x} \]

   2. lift-log.f64 N/A

      \[\leadsto \log x - {\color{blue}{\log \log x}}^{2} \cdot \frac{1}{\log \log x} \]

   3. metadata-eval N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\log \log x} \]

   4. lift-log.f64 N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \color{blue}{\log x}} \]

   5. unpow1 N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \color{blue}{\left({\log x}^{1}\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left({\log x}^{\color{blue}{\left(2 - 1\right)}}\right)} \]

   7. pow-div N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \color{blue}{\left(\frac{{\log x}^{2}}{{\log x}^{1}}\right)}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left(\frac{\color{blue}{{\log x}^{2}}}{{\log x}^{1}}\right)} \]

   9. unpow1 N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left(\frac{{\log x}^{2}}{\color{blue}{\log x}}\right)} \]

   10. clear-num N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \color{blue}{\left(\frac{1}{\frac{\log x}{{\log x}^{2}}}\right)}} \]

   11. --rgt-identity N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left(\frac{1}{\frac{\log x}{\color{blue}{{\log x}^{2} - 0}}}\right)} \]

   12. lift--.f64 N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left(\frac{1}{\frac{\log x}{\color{blue}{{\log x}^{2} - 0}}}\right)} \]

   13. log-rec N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\mathsf{neg}\left(\log \left(\frac{\log x}{{\log x}^{2} - 0}\right)\right)}} \]

   14. unpow1 N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left(\frac{\color{blue}{{\log x}^{1}}}{{\log x}^{2} - 0}\right)\right)} \]

   15. lift--.f64 N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left(\frac{{\log x}^{1}}{\color{blue}{{\log x}^{2} - 0}}\right)\right)} \]

   16. --rgt-identity N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left(\frac{{\log x}^{1}}{\color{blue}{{\log x}^{2}}}\right)\right)} \]

   17. lift-pow.f64 N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left(\frac{{\log x}^{1}}{\color{blue}{{\log x}^{2}}}\right)\right)} \]

   18. pow-div N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \color{blue}{\left({\log x}^{\left(1 - 2\right)}\right)}\right)} \]

   19. metadata-eval N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left({\log x}^{\color{blue}{-1}}\right)\right)} \]

   20. inv-pow N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \color{blue}{\left(\frac{1}{\log x}\right)}\right)} \]

6. Applied rewrites 100.0%

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

8. Applied rewrites 30.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. frac-2neg N/A

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

   7. *-rgt-identity N/A

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

   8. log-div N/A

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

   9. *-rgt-identity N/A

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

   10. log-div N/A

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

   11. frac-2neg N/A

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

   12. clear-num N/A

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

   13. log-rec N/A

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

   14. lower-neg.f64 N/A

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

10. Applied rewrites 100.0%

    \[\leadsto \color{blue}{-\log \left(\frac{\log x}{x}\right)} \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-log.f64 N/A

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

   2. lift-log.f64 99.6

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

   3. remove-double-neg N/A

      \[\leadsto \log x - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \log x\right)\right)\right)\right)} \]

   4. neg-sub0 N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\color{blue}{\left(0 - \log \log x\right)}\right)\right) \]

   5. flip3-- N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\color{blue}{\frac{{0}^{3} - {\log \log x}^{3}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{0} - {\log \log x}^{3}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   7. neg-sub0 N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({\log \log x}^{3}\right)}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   8. cube-neg N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(\log \log x\right)\right)}^{3}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   9. sqr-pow N/A

      \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\mathsf{neg}\left(\log \log x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(\log \log x\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   10. unpow-prod-down N/A

       \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(\log \log x\right)\right) \cdot \left(\mathsf{neg}\left(\log \log x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   11. sqr-neg N/A

       \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(\log \log x \cdot \log \log x\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   12. pow-prod-down N/A

       \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\log \log x}^{\left(\frac{3}{2}\right)} \cdot {\log \log x}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   13. sqr-pow N/A

       \[\leadsto \log x - \left(\mathsf{neg}\left(\frac{\color{blue}{{\log \log x}^{3}}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)\right) \]

   14. distribute-frac-neg N/A

       \[\leadsto \log x - \color{blue}{\frac{\mathsf{neg}\left({\log \log x}^{3}\right)}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}} \]

4. Applied rewrites 99.6%

   \[\leadsto \log x - \color{blue}{{\log \log x}^{2} \cdot \frac{1}{\log \log x}} \]
5. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \log x - {\log \color{blue}{\log x}}^{2} \cdot \frac{1}{\log \log x} \]

   2. lift-log.f64 N/A

      \[\leadsto \log x - {\color{blue}{\log \log x}}^{2} \cdot \frac{1}{\log \log x} \]

   3. metadata-eval N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\color{blue}{\mathsf{neg}\left(-1\right)}}{\log \log x} \]

   4. lift-log.f64 N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \color{blue}{\log x}} \]

   5. unpow1 N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \color{blue}{\left({\log x}^{1}\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left({\log x}^{\color{blue}{\left(2 - 1\right)}}\right)} \]

   7. pow-div N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \color{blue}{\left(\frac{{\log x}^{2}}{{\log x}^{1}}\right)}} \]

   8. lift-pow.f64 N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left(\frac{\color{blue}{{\log x}^{2}}}{{\log x}^{1}}\right)} \]

   9. unpow1 N/A

      \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left(\frac{{\log x}^{2}}{\color{blue}{\log x}}\right)} \]

   10. clear-num N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \color{blue}{\left(\frac{1}{\frac{\log x}{{\log x}^{2}}}\right)}} \]

   11. --rgt-identity N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left(\frac{1}{\frac{\log x}{\color{blue}{{\log x}^{2} - 0}}}\right)} \]

   12. lift--.f64 N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\log \left(\frac{1}{\frac{\log x}{\color{blue}{{\log x}^{2} - 0}}}\right)} \]

   13. log-rec N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\mathsf{neg}\left(\log \left(\frac{\log x}{{\log x}^{2} - 0}\right)\right)}} \]

   14. unpow1 N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left(\frac{\color{blue}{{\log x}^{1}}}{{\log x}^{2} - 0}\right)\right)} \]

   15. lift--.f64 N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left(\frac{{\log x}^{1}}{\color{blue}{{\log x}^{2} - 0}}\right)\right)} \]

   16. --rgt-identity N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left(\frac{{\log x}^{1}}{\color{blue}{{\log x}^{2}}}\right)\right)} \]

   17. lift-pow.f64 N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left(\frac{{\log x}^{1}}{\color{blue}{{\log x}^{2}}}\right)\right)} \]

   18. pow-div N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \color{blue}{\left({\log x}^{\left(1 - 2\right)}\right)}\right)} \]

   19. metadata-eval N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \left({\log x}^{\color{blue}{-1}}\right)\right)} \]

   20. inv-pow N/A

       \[\leadsto \log x - {\log \log x}^{2} \cdot \frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\log \color{blue}{\left(\frac{1}{\log x}\right)}\right)} \]

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\log \left(\frac{x}{\log x}\right)} \]
7. Add Preprocessing

Alternative 3: 24.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-log.f64 N/A

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

   2. diff-log N/A

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

   3. lower-log.f64 N/A

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

   4. div-inv N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

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

   7. lift-log.f64 N/A

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

   8. *-commutative N/A

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

   9. neg-mul-1 N/A

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

   10. neg-sub0 N/A

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

   11. flip3-- N/A

       \[\leadsto \log \left(x \cdot e^{\color{blue}{\frac{{0}^{3} - {\log \log x}^{3}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \log \left(x \cdot e^{\frac{\color{blue}{0} - {\log \log x}^{3}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}}\right) \]

   13. neg-sub0 N/A

       \[\leadsto \log \left(x \cdot e^{\frac{\color{blue}{\mathsf{neg}\left({\log \log x}^{3}\right)}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}}\right) \]

   14. distribute-frac-neg N/A

       \[\leadsto \log \left(x \cdot e^{\color{blue}{\mathsf{neg}\left(\frac{{\log \log x}^{3}}{0 \cdot 0 + \left(\log \log x \cdot \log \log x + 0 \cdot \log \log x\right)}\right)}}\right) \]

4. Applied rewrites 25.0%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))