Average Error: 31.8 → 0.1
Time: 4.9s
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 r89378 = x;
        double r89379 = r89378 * r89378;
        double r89380 = 1.0;
        double r89381 = r89379 - r89380;
        double r89382 = sqrt(r89381);
        double r89383 = r89378 + r89382;
        double r89384 = log(r89383);
        return r89384;
}
↓
double f(double x) {
        double r89385 = x;
        double r89386 = 1.0;
        double r89387 = sqrt(r89386);
        double r89388 = r89385 + r89387;
        double r89389 = sqrt(r89388);
        double r89390 = r89385 - r89387;
        double r89391 = sqrt(r89390);
        double r89392 = r89389 * r89391;
        double r89393 = r89385 + r89392;
        double r89394 = log(r89393);
        return r89394;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.8

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

  3. Applied add-sqr-sqrt31.8

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

  4. Applied difference-of-squares31.8

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

  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 2020001 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))