Average Error: 32.6 → 0.1
Time: 3.3s
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 r43469 = x;
        double r43470 = r43469 * r43469;
        double r43471 = 1.0;
        double r43472 = r43470 - r43471;
        double r43473 = sqrt(r43472);
        double r43474 = r43469 + r43473;
        double r43475 = log(r43474);
        return r43475;
}
↓
double f(double x) {
        double r43476 = x;
        double r43477 = 1.0;
        double r43478 = sqrt(r43477);
        double r43479 = r43476 + r43478;
        double r43480 = sqrt(r43479);
        double r43481 = r43476 - r43478;
        double r43482 = sqrt(r43481);
        double r43483 = r43480 * r43482;
        double r43484 = r43476 + r43483;
        double r43485 = log(r43484);
        return r43485;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  3. Applied add-sqr-sqrt32.6

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

  4. Applied difference-of-squares32.6

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

  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 2020034 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))