Average Error: 32.5 → 0.1
Time: 2.5s
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 r59873 = x;
        double r59874 = r59873 * r59873;
        double r59875 = 1.0;
        double r59876 = r59874 - r59875;
        double r59877 = sqrt(r59876);
        double r59878 = r59873 + r59877;
        double r59879 = log(r59878);
        return r59879;
}
↓
double f(double x) {
        double r59880 = x;
        double r59881 = 1.0;
        double r59882 = sqrt(r59881);
        double r59883 = r59880 + r59882;
        double r59884 = sqrt(r59883);
        double r59885 = r59880 - r59882;
        double r59886 = sqrt(r59885);
        double r59887 = r59884 * r59886;
        double r59888 = r59880 + r59887;
        double r59889 = log(r59888);
        return r59889;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.5

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

  3. Applied add-sqr-sqrt32.5

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

  4. Applied difference-of-squares32.5

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

  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 2020025 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))