Average Error: 31.5 → 0.1
Time: 3.9s
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 r82219 = x;
        double r82220 = r82219 * r82219;
        double r82221 = 1.0;
        double r82222 = r82220 - r82221;
        double r82223 = sqrt(r82222);
        double r82224 = r82219 + r82223;
        double r82225 = log(r82224);
        return r82225;
}
↓
double f(double x) {
        double r82226 = x;
        double r82227 = 1.0;
        double r82228 = sqrt(r82227);
        double r82229 = r82226 + r82228;
        double r82230 = sqrt(r82229);
        double r82231 = r82226 - r82228;
        double r82232 = sqrt(r82231);
        double r82233 = r82230 * r82232;
        double r82234 = r82226 + r82233;
        double r82235 = log(r82234);
        return r82235;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  3. Applied add-sqr-sqrt31.5

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

  4. Applied difference-of-squares31.5

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

  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 2020056 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))