Average Error: 32.6 → 0.1
Time: 19.1s
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 r70161 = x;
        double r70162 = r70161 * r70161;
        double r70163 = 1.0;
        double r70164 = r70162 - r70163;
        double r70165 = sqrt(r70164);
        double r70166 = r70161 + r70165;
        double r70167 = log(r70166);
        return r70167;
}
↓
double f(double x) {
        double r70168 = x;
        double r70169 = 1.0;
        double r70170 = sqrt(r70169);
        double r70171 = r70168 + r70170;
        double r70172 = sqrt(r70171);
        double r70173 = r70168 - r70170;
        double r70174 = sqrt(r70173);
        double r70175 = r70172 * r70174;
        double r70176 = r70168 + r70175;
        double r70177 = log(r70176);
        return r70177;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  3. Applied add-sqr-sqrt32.6

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

  4. Applied difference-of-squares32.6

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

  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 2019326 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))