Jmat.Real.lambertw, estimator

Percentage Accurate: 99.6% → 100.0%

Time: 4.0s

Alternatives: 2

Speedup: 1.5×

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, 1.5× 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 99.5%

   \[\log x - \log \log x \]

2. Taylor expanded in x around inf 99.5%

   \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) - \log \left(-1 \cdot \log \left(\frac{1}{x}\right)\right)} \]
3. Step-by-step derivation

   1. mul-1-neg 99.5%

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

   2. log-rec 99.5%

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

   3. remove-double-neg 99.5%

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

   4. mul-1-neg 99.5%

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

   5. log-rec 99.5%

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

   6. remove-double-neg 99.5%

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

   7. log-div 100.0%

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

4. Simplified 100.0%

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

   1. clear-num 100.0%

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

   2. log-rec 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.5× 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 99.5%

   \[\log x - \log \log x \]

2. Taylor expanded in x around inf 99.5%

   \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{x}\right) - \log \left(-1 \cdot \log \left(\frac{1}{x}\right)\right)} \]
3. Step-by-step derivation

   1. mul-1-neg 99.5%

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

   2. log-rec 99.5%

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

   3. remove-double-neg 99.5%

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

   4. mul-1-neg 99.5%

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

   5. log-rec 99.5%

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

   6. remove-double-neg 99.5%

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

   7. log-div 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2023182 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))