Average Error: 32.4 → 0.1
Time: 7.7s
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 r32983 = x;
        double r32984 = r32983 * r32983;
        double r32985 = 1.0;
        double r32986 = r32984 - r32985;
        double r32987 = sqrt(r32986);
        double r32988 = r32983 + r32987;
        double r32989 = log(r32988);
        return r32989;
}
↓
double f(double x) {
        double r32990 = x;
        double r32991 = 1.0;
        double r32992 = sqrt(r32991);
        double r32993 = r32990 - r32992;
        double r32994 = sqrt(r32993);
        double r32995 = r32990 + r32992;
        double r32996 = sqrt(r32995);
        double r32997 = r32994 * r32996;
        double r32998 = r32990 + r32997;
        double r32999 = log(r32998);
        return r32999;
}

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. Simplified0.1

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

  7. 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)))))