Average Error: 32.1 → 0.1
Time: 4.9s
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 r55184 = x;
        double r55185 = r55184 * r55184;
        double r55186 = 1.0;
        double r55187 = r55185 - r55186;
        double r55188 = sqrt(r55187);
        double r55189 = r55184 + r55188;
        double r55190 = log(r55189);
        return r55190;
}
↓
double f(double x) {
        double r55191 = x;
        double r55192 = 1.0;
        double r55193 = sqrt(r55192);
        double r55194 = r55191 + r55193;
        double r55195 = sqrt(r55194);
        double r55196 = r55191 - r55193;
        double r55197 = sqrt(r55196);
        double r55198 = r55195 * r55197;
        double r55199 = r55191 + r55198;
        double r55200 = log(r55199);
        return r55200;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.1

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

  3. Applied add-sqr-sqrt32.1

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

  4. Applied difference-of-squares32.1

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

  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 2020042 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))