Average Error: 32.2 → 0.3
Time: 2.5s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right)\right) \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x) :precision binary64 (log (fma x 2.0 (/ -0.5 x))))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log(fma(x, 2.0, (-0.5 / x)));
}

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end
↓
function code(x)
	return log(fma(x, 2.0, Float64(-0.5 / x)))
end

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]
↓
code[x_] := N[Log[N[(x * 2.0 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right)\right)

Error

Bits error versus x

Derivation

  1. Initial program 32.2

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

  2. Taylor expanded in x around inf 0.3

    \[\leadsto \log \color{blue}{\left(2 \cdot x - 0.5 \cdot \frac{1}{x}\right)} \]

  3. Simplified0.3

    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right)\right)} \]

  4. Final simplification0.3

    \[\leadsto \log \left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right)\right) \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))