Average Error: 32.2 → 0.2
Time: 3.4s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(\left(\frac{-0.5}{x} + \frac{-0.125}{{x}^{3}}\right) + \mathsf{fma}\left(x, 2, \frac{-0.0625}{{x}^{5}}\right)\right) \]
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (log
  (+
   (+ (/ -0.5 x) (/ -0.125 (pow x 3.0)))
   (fma x 2.0 (/ -0.0625 (pow x 5.0))))))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log((((-0.5 / x) + (-0.125 / pow(x, 3.0))) + fma(x, 2.0, (-0.0625 / pow(x, 5.0)))));
}

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end
↓
function code(x)
	return log(Float64(Float64(Float64(-0.5 / x) + Float64(-0.125 / (x ^ 3.0))) + fma(x, 2.0, Float64(-0.0625 / (x ^ 5.0)))))
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[(N[(N[(-0.5 / x), $MachinePrecision] + N[(-0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0 + N[(-0.0625 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\left(\frac{-0.5}{x} + \frac{-0.125}{{x}^{3}}\right) + \mathsf{fma}\left(x, 2, \frac{-0.0625}{{x}^{5}}\right)\right)

Error

Derivation

  1. Initial program 32.2

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

  2. Simplified32.2

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

  3. Taylor expanded in x around inf 0.2

    \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + \left(0.0625 \cdot \frac{1}{{x}^{5}} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

  4. Simplified0.2

    \[\leadsto \log \color{blue}{\left(\left(\frac{-0.5}{x} + \frac{-0.125}{{x}^{3}}\right) + \mathsf{fma}\left(x, 2, \frac{-0.0625}{{x}^{5}}\right)\right)} \]

  5. Final simplification0.2

    \[\leadsto \log \left(\left(\frac{-0.5}{x} + \frac{-0.125}{{x}^{3}}\right) + \mathsf{fma}\left(x, 2, \frac{-0.0625}{{x}^{5}}\right)\right) \]

Reproduce

herbie shell --seed 2022209 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))