Jmat.Real.lambertw, estimator

Average Error: 0.3 → 0.0

Time: 2.8s

Precision: binary64

Cost: 13120

Math

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

↓

\[\mathsf{log1p}\left(\frac{x}{\log x} + -1\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (log1p (+ (/ x (log x)) -1.0)))

C

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

↓

double code(double x) {
	return log1p(((x / log(x)) + -1.0));
}

Java

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

↓

public static double code(double x) {
	return Math.log1p(((x / Math.log(x)) + -1.0));
}

Python

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

↓

def code(x):
	return math.log1p(((x / math.log(x)) + -1.0))

Julia

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

↓

function code(x)
	return log1p(Float64(Float64(x / log(x)) + -1.0))
end

Wolfram

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

↓

code[x_] := N[Log[1 + N[(N[(x / N[Log[x], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.3

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

2. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{x}{\log x} - 1\right)} \]

3. Final simplification 0.0

   \[\leadsto \mathsf{log1p}\left(\frac{x}{\log x} + -1\right) \]

Alternatives

Alternative 1
Error	0.0
Cost	12992

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

Reproduce

herbie shell --seed 2022298 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))