Jmat.Real.lambertw, estimator

Percentage Accurate: 99.6% → 100.0%

Time: 2.7s

Alternatives: 2

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--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. neg-sub0 N/A

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

   5. associate-+l- N/A

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

   6. neg-sub0 N/A

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

   7. lower-neg.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. diff-log N/A

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

   11. lower-log.f64 N/A

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

   12. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-neg.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. neg-log N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lower-log.f64 N/A

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

   7. lower-/.f64 100.0

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

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

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

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 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--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. diff-log N/A

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

   5. lower-log.f64 N/A

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

   6. div-inv N/A

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

   7. inv-pow N/A

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

   8. pow-to-exp N/A

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

   9. lift-log.f64 N/A

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

   10. *-commutative N/A

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

   11. neg-mul-1 N/A

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

   12. neg-sub0 N/A

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

   13. flip3-- N/A

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

   14. metadata-eval N/A

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

   15. neg-sub0 N/A

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

   16. 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 24.8%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))