Average Error: 30.9 → 0.1
Time: 14.1s
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 r1060385 = x;
        double r1060386 = r1060385 * r1060385;
        double r1060387 = 1.0;
        double r1060388 = r1060386 - r1060387;
        double r1060389 = sqrt(r1060388);
        double r1060390 = r1060385 + r1060389;
        double r1060391 = log(r1060390);
        return r1060391;
}
↓
double f(double x) {
        double r1060392 = x;
        double r1060393 = 1.0;
        double r1060394 = r1060393 + r1060392;
        double r1060395 = sqrt(r1060394);
        double r1060396 = r1060392 - r1060393;
        double r1060397 = sqrt(r1060396);
        double r1060398 = r1060395 * r1060397;
        double r1060399 = r1060392 + r1060398;
        double r1060400 = log(r1060399);
        return r1060400;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

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

  3. Applied *-un-lft-identity30.9

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

  4. Applied difference-of-squares30.9

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

  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 2019153 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))