Average Error: 30.9 → 0.1
Time: 11.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right) + \log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right) + \log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right)
double f(double x) {
        double r1171042 = x;
        double r1171043 = r1171042 * r1171042;
        double r1171044 = 1.0;
        double r1171045 = r1171043 - r1171044;
        double r1171046 = sqrt(r1171045);
        double r1171047 = r1171042 + r1171046;
        double r1171048 = log(r1171047);
        return r1171048;
}
↓
double f(double x) {
        double r1171049 = x;
        double r1171050 = 1.0;
        double r1171051 = r1171050 + r1171049;
        double r1171052 = sqrt(r1171051);
        double r1171053 = r1171049 - r1171050;
        double r1171054 = sqrt(r1171053);
        double r1171055 = r1171052 * r1171054;
        double r1171056 = r1171049 + r1171055;
        double r1171057 = sqrt(r1171056);
        double r1171058 = log(r1171057);
        double r1171059 = r1171058 + r1171058;
        return r1171059;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

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

  3. Applied *-un-lft-identity30.9

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

  4. Applied difference-of-squares30.9

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied log-prod0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))