Average Error: 31.5 → 0.1
Time: 5.6s
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 r84183 = x;
        double r84184 = r84183 * r84183;
        double r84185 = 1.0;
        double r84186 = r84184 - r84185;
        double r84187 = sqrt(r84186);
        double r84188 = r84183 + r84187;
        double r84189 = log(r84188);
        return r84189;
}
↓
double f(double x) {
        double r84190 = x;
        double r84191 = 1.0;
        double r84192 = sqrt(r84191);
        double r84193 = r84190 + r84192;
        double r84194 = sqrt(r84193);
        double r84195 = r84190 - r84192;
        double r84196 = sqrt(r84195);
        double r84197 = r84194 * r84196;
        double r84198 = r84190 + r84197;
        double r84199 = log(r84198);
        return r84199;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  3. Applied add-sqr-sqrt31.5

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

  4. Applied difference-of-squares31.5

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

  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 2020056 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))