Average Error: 31.5 → 0.1
Time: 11.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)
double f(double x) {
        double r1733379 = x;
        double r1733380 = r1733379 * r1733379;
        double r1733381 = 1.0;
        double r1733382 = r1733380 - r1733381;
        double r1733383 = sqrt(r1733382);
        double r1733384 = r1733379 + r1733383;
        double r1733385 = log(r1733384);
        return r1733385;
}
↓
double f(double x) {
        double r1733386 = x;
        double r1733387 = 1.0;
        double r1733388 = r1733387 + r1733386;
        double r1733389 = sqrt(r1733388);
        double r1733390 = r1733386 - r1733387;
        double r1733391 = sqrt(r1733390);
        double r1733392 = r1733389 * r1733391;
        double r1733393 = r1733386 + r1733392;
        double r1733394 = log(r1733393);
        return r1733394;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity31.5

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

  4. Applied difference-of-squares31.5

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019133 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))