Average Error: 32.0 → 0.1
Time: 9.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x - \sqrt{1}} \cdot \sqrt{x + \sqrt{1}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x - \sqrt{1}} \cdot \sqrt{x + \sqrt{1}}\right)
double f(double x) {
        double r1508571 = x;
        double r1508572 = r1508571 * r1508571;
        double r1508573 = 1.0;
        double r1508574 = r1508572 - r1508573;
        double r1508575 = sqrt(r1508574);
        double r1508576 = r1508571 + r1508575;
        double r1508577 = log(r1508576);
        return r1508577;
}
↓
double f(double x) {
        double r1508578 = x;
        double r1508579 = 1.0;
        double r1508580 = sqrt(r1508579);
        double r1508581 = r1508578 - r1508580;
        double r1508582 = sqrt(r1508581);
        double r1508583 = r1508578 + r1508580;
        double r1508584 = sqrt(r1508583);
        double r1508585 = r1508582 * r1508584;
        double r1508586 = r1508578 + r1508585;
        double r1508587 = log(r1508586);
        return r1508587;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 32.0

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

  3. Applied add-sqr-sqrt32.0

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

  4. Applied difference-of-squares32.0

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019169 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))