Jmat.Real.lambertw, estimator

Average Error: 0.3 → 0.0

Time: 2.0s

Precision: binary64

Cost: 12992

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return log((x / 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((x / 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((x / Math.log(x)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.3

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

2. Applied egg-rr 0.3

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

3. Simplified 0.0

   \[\leadsto \color{blue}{\log \left(\frac{x}{\log x}\right)} \]
   Proof
   (log.f64 (/.f64 x (log.f64 x))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> log-div_binary64 (-.f64 (log.f64 x) (log.f64 (log.f64 x)))): 72 points increase in error, 1 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 (log.f64 x) (neg.f64 (log.f64 (log.f64 x))))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.0

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

Reproduce

herbie shell --seed 2022334 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))