Jmat.Real.lambertw, estimator

Average Error: 0.3 → 0.3

Time: 1.6s

Precision: binary64

Cost: 19584

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.3

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

2. Taylor expanded in x around inf 0.3

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

3. Taylor expanded in x around 0 0.3

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

4. Applied egg-rr 0.3

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

5. Simplified 0.3

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

   [Start] 0.3	\[ \left(-\log \left(\frac{1}{x}\right)\right) + \left(-\log \log x\right) \]
   rational.json-simplify-1 [<=] 0.3	\[ \color{blue}{\left(-\log \log x\right) + \left(-\log \left(\frac{1}{x}\right)\right)} \]
   rational.json-simplify-41 [<=] 0.3	\[ \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right) - \log \log x} \]

6. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	19392

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

Reproduce

herbie shell --seed 2023053 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))