?

Average Error: 0.4 → 0.1
Time: 10.5s
Precision: binary64
Cost: 19264

?

\[-1000 \leq x \land x \leq 0\]
\[\log \log \left(1 + e^{x}\right) \]
↓
\[\log \left(\mathsf{log1p}\left(e^{x}\right)\right) \]
(FPCore (x) :precision binary64 (log (log (+ 1.0 (exp x)))))
↓
(FPCore (x) :precision binary64 (log (log1p (exp x))))
double code(double x) {
	return log(log((1.0 + exp(x))));
}
↓
double code(double x) {
	return log(log1p(exp(x)));
}

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

def code(x):
	return math.log(math.log((1.0 + math.exp(x))))
↓
def code(x):
	return math.log(math.log1p(math.exp(x)))

function code(x)
	return log(log(Float64(1.0 + exp(x))))
end
↓
function code(x)
	return log(log1p(exp(x)))
end

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

\log \log \left(1 + e^{x}\right)
↓
\log \left(\mathsf{log1p}\left(e^{x}\right)\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.4

    \[\log \log \left(1 + e^{x}\right) \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\log \left(\mathsf{log1p}\left(e^{x}\right)\right)} \]
    Proof

    [Start]0.4

    \[ \log \log \left(1 + e^{x}\right) \]

    log1p-def [=>]0.1

    \[ \log \color{blue}{\left(\mathsf{log1p}\left(e^{x}\right)\right)} \]

  3. Final simplification0.1

    \[\leadsto \log \left(\mathsf{log1p}\left(e^{x}\right)\right) \]

Alternatives

Alternative 1
Error1.3
Cost13120
\[\log \left(x \cdot 0.5 + \log 2\right) \]
Alternative 2
Error1.4
Cost12992
\[\log \log \left(x + 2\right) \]
Alternative 3
Error2.0
Cost12864
\[\log \log 2 \]
Alternative 4
Error64.0
Cost6592
\[\log \left(x \cdot 0.5\right) \]

Error

Reproduce?

herbie shell --seed 1 
(FPCore (x)
  :name "log(log(1+exp(x)))"
  :precision binary64
  :pre (and (<= -1000.0 x) (<= x 0.0))
  (log (log (+ 1.0 (exp x)))))