Average Error: 31.3 → 0.8
Time: 4.7s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log x + \log 2 \]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log x + \log 2

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x) :precision binary64 (+ (log x) (log 2.0)))
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log(x) + log(2.0);
}

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 pow1_binary640.8

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

  6. Applied log-pow_binary640.8

    \[\leadsto \log 2 + \color{blue}{1 \cdot \log x} \]

  7. Applied pow1_binary640.8

    \[\leadsto \log \color{blue}{\left({2}^{1}\right)} + 1 \cdot \log x \]

  8. Applied log-pow_binary640.8

    \[\leadsto \color{blue}{1 \cdot \log 2} + 1 \cdot \log x \]

  9. Applied distribute-lft-out_binary640.8

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

  10. Simplified0.8

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

  11. Final simplification0.8

    \[\leadsto \log x + \log 2 \]

Reproduce

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