Jmat.Real.lambertw, estimator

Percentage Accurate: 99.6% → 100.0%

Time: 5.0s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. remove-double-neg N/A

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

   2. neg-sub0 N/A

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

   3. flip3-- N/A

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

   4. metadata-eval N/A

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

   5. neg-sub0 N/A

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

   6. cube-neg N/A

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

   7. sqr-pow N/A

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

   8. unpow-prod-down N/A

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

   9. sqr-neg N/A

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

   10. pow-prod-down N/A

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

   11. sqr-pow N/A

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

   12. 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 egg-rr 99.6%

   \[\leadsto \log x - \color{blue}{\frac{1}{\frac{{\log \log x}^{2} + 0}{{\log \log x}^{3}}}} \]
5. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \log x - \frac{1}{\frac{\color{blue}{{\log \log x}^{2}}}{{\log \log x}^{3}}} \]

   2. clear-num N/A

      \[\leadsto \log x - \color{blue}{\frac{{\log \log x}^{3}}{{\log \log x}^{2}}} \]

   3. pow-div N/A

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

   4. metadata-eval N/A

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

   5. unpow1 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. neg-log N/A

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

   9. sum-log N/A

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

   10. log-lowering-log.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. /-lowering-/.f64 N/A

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

   13. log-lowering-log.f64 100.0

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \log \left(x \cdot \frac{1}{\log x}\right) \]
8. 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. remove-double-neg N/A

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

   2. neg-sub0 N/A

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

   3. flip3-- N/A

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

   4. metadata-eval N/A

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

   5. neg-sub0 N/A

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

   6. cube-neg N/A

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

   7. sqr-pow N/A

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

   8. unpow-prod-down N/A

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

   9. sqr-neg N/A

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

   10. pow-prod-down N/A

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

   11. sqr-pow N/A

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

   12. 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 egg-rr 99.6%

   \[\leadsto \log x - \color{blue}{\frac{1}{\frac{{\log \log x}^{2} + 0}{{\log \log x}^{3}}}} \]
5. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \log x - \frac{1}{\frac{\color{blue}{{\log \log x}^{2}}}{{\log \log x}^{3}}} \]

   2. clear-num N/A

      \[\leadsto \log x - \color{blue}{\frac{{\log \log x}^{3}}{{\log \log x}^{2}}} \]

   3. pow-div N/A

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

   4. metadata-eval N/A

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

   5. unpow1 N/A

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

   6. diff-log N/A

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

   7. log-lowering-log.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. log-lowering-log.f64 100.0

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

6. Applied egg-rr 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. diff-log N/A

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

   2. log-lowering-log.f64 N/A

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

   3. div-inv N/A

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

   4. inv-pow N/A

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

   5. pow-to-exp N/A

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

   6. *-commutative N/A

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

   7. neg-mul-1 N/A

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

   8. neg-sub0 N/A

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

   9. flip3-- N/A

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

   10. metadata-eval N/A

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

   11. neg-sub0 N/A

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

   12. 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 egg-rr 25.0%

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

Reproduce

herbie shell --seed 2024205 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))