Average Error: 32.0 → 0.0
Time: 8.6s
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 r28279 = x;
        double r28280 = r28279 * r28279;
        double r28281 = 1.0;
        double r28282 = r28280 - r28281;
        double r28283 = sqrt(r28282);
        double r28284 = r28279 + r28283;
        double r28285 = log(r28284);
        return r28285;
}
↓
double f(double x) {
        double r28286 = x;
        double r28287 = 1.0;
        double r28288 = sqrt(r28287);
        double r28289 = r28286 + r28288;
        double r28290 = sqrt(r28289);
        double r28291 = r28286 - r28288;
        double r28292 = sqrt(r28291);
        double r28293 = r28290 * r28292;
        double r28294 = r28286 + r28293;
        double r28295 = log(r28294);
        return r28295;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))