Average Error: 32.4 → 0.1
Time: 9.0s
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 r34252 = x;
        double r34253 = r34252 * r34252;
        double r34254 = 1.0;
        double r34255 = r34253 - r34254;
        double r34256 = sqrt(r34255);
        double r34257 = r34252 + r34256;
        double r34258 = log(r34257);
        return r34258;
}
↓
double f(double x) {
        double r34259 = x;
        double r34260 = 1.0;
        double r34261 = sqrt(r34260);
        double r34262 = r34259 - r34261;
        double r34263 = sqrt(r34262);
        double r34264 = r34259 + r34261;
        double r34265 = sqrt(r34264);
        double r34266 = r34263 * r34265;
        double r34267 = r34259 + r34266;
        double r34268 = log(r34267);
        return r34268;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 32.4

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

  3. Applied add-sqr-sqrt32.4

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

  4. Applied difference-of-squares32.4

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

  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 2019179 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))