Jmat.Real.lambertw, estimator

Percentage Accurate: 99.6% → 100.0%

Time: 4.4s

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

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

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 99.6

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

   2. unpow1 N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 99.6

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

4. Applied rewrites 99.6%

   \[\leadsto \log x - \log \color{blue}{\left({\log x}^{0.5} \cdot {\log x}^{0.5}\right)} \]
5. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. pow-prod-up N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. unpow1 N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-up N/A

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

   12. lift-pow.f64 N/A

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

   13. lift-pow.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. unpow-prod-down N/A

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

6. Applied rewrites 100.0%

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

Alternative 2: 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 25.1%

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

Alternative 3: 0.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \log x + \frac{0}{0} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (log x) (/ 0.0 0.0)))

C

double code(double x) {
	return log(x) + (0.0 / 0.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(x) + (0.0d0 / 0.0d0)
end function

Java

public static double code(double x) {
	return Math.log(x) + (0.0 / 0.0);
}

Python

def code(x):
	return math.log(x) + (0.0 / 0.0)

Julia

function code(x)
	return Float64(log(x) + Float64(0.0 / 0.0))
end

MATLAB

function tmp = code(x)
	tmp = log(x) + (0.0 / 0.0);
end

Wolfram

code[x_] := N[(N[Log[x], $MachinePrecision] + N[(0.0 / 0.0), $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 99.6

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

   2. unpow1 N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 99.6

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

4. Applied rewrites 99.6%

   \[\leadsto \log x - \log \color{blue}{\left({\log x}^{0.5} \cdot {\log x}^{0.5}\right)} \]
5. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. unpow1 N/A

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

   4. metadata-eval N/A

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

   5. pow-prod-up N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. unpow1 N/A

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

   11. metadata-eval N/A

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

   12. pow-prod-up N/A

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

   13. lift-pow.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. unpow-prod-down N/A

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

6. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\log x - \frac{0}{0}} \]

7. Final simplification 0.0%

   \[\leadsto \log x + \frac{0}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))