Average Error: 32.4 → 0.1
Time: 8.3s
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 r3094209 = x;
        double r3094210 = r3094209 * r3094209;
        double r3094211 = 1.0;
        double r3094212 = r3094210 - r3094211;
        double r3094213 = sqrt(r3094212);
        double r3094214 = r3094209 + r3094213;
        double r3094215 = log(r3094214);
        return r3094215;
}
↓
double f(double x) {
        double r3094216 = x;
        double r3094217 = 1.0;
        double r3094218 = sqrt(r3094217);
        double r3094219 = r3094216 - r3094218;
        double r3094220 = sqrt(r3094219);
        double r3094221 = r3094216 + r3094218;
        double r3094222 = sqrt(r3094221);
        double r3094223 = r3094220 * r3094222;
        double r3094224 = r3094216 + r3094223;
        double r3094225 = log(r3094224);
        return r3094225;
}

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