Average Error: 31.7 → 0.1
Time: 5.3s
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 r80182 = x;
        double r80183 = r80182 * r80182;
        double r80184 = 1.0;
        double r80185 = r80183 - r80184;
        double r80186 = sqrt(r80185);
        double r80187 = r80182 + r80186;
        double r80188 = log(r80187);
        return r80188;
}
↓
double f(double x) {
        double r80189 = x;
        double r80190 = 1.0;
        double r80191 = sqrt(r80190);
        double r80192 = r80189 + r80191;
        double r80193 = sqrt(r80192);
        double r80194 = r80189 - r80191;
        double r80195 = sqrt(r80194);
        double r80196 = r80193 * r80195;
        double r80197 = r80189 + r80196;
        double r80198 = log(r80197);
        return r80198;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  3. Applied add-sqr-sqrt31.7

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

  4. Applied difference-of-squares31.7

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

  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 2020064 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))