Average Error: 31.3 → 0.8
Time: 3.2s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\mathsf{log1p}\left(1\right) + \log x \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x) :precision binary64 (+ (log1p 1.0) (log x)))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log1p(1.0) + log(x);
}

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

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))
↓
def code(x):
	return math.log1p(1.0) + math.log(x)

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end
↓
function code(x)
	return Float64(log1p(1.0) + log(x))
end

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

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\mathsf{log1p}\left(1\right) + \log x

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

    \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]

  2. Simplified31.3

    \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)} \]

  3. Taylor expanded in x around inf 0.8

    \[\leadsto \color{blue}{\log 2 - \log \left(\frac{1}{x}\right)} \]

  4. Simplified0.8

    \[\leadsto \color{blue}{\log 2 + \log x} \]

  5. Applied egg-rr1.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)} \]

  6. Applied egg-rr0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2022160 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))