Average Error: 31.6 → 0.1
Time: 2.7s
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 r43122 = x;
        double r43123 = r43122 * r43122;
        double r43124 = 1.0;
        double r43125 = r43123 - r43124;
        double r43126 = sqrt(r43125);
        double r43127 = r43122 + r43126;
        double r43128 = log(r43127);
        return r43128;
}
↓
double f(double x) {
        double r43129 = x;
        double r43130 = 1.0;
        double r43131 = sqrt(r43130);
        double r43132 = r43129 + r43131;
        double r43133 = sqrt(r43132);
        double r43134 = r43129 - r43131;
        double r43135 = sqrt(r43134);
        double r43136 = r43133 * r43135;
        double r43137 = r43129 + r43136;
        double r43138 = log(r43137);
        return r43138;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.6

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

  3. Applied add-sqr-sqrt31.6

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

  4. Applied difference-of-squares31.6

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

  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 2019352 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))