Average Error: 32.2 → 0.1
Time: 7.9s
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 r83213 = x;
        double r83214 = r83213 * r83213;
        double r83215 = 1.0;
        double r83216 = r83214 - r83215;
        double r83217 = sqrt(r83216);
        double r83218 = r83213 + r83217;
        double r83219 = log(r83218);
        return r83219;
}
↓
double f(double x) {
        double r83220 = x;
        double r83221 = 1.0;
        double r83222 = sqrt(r83221);
        double r83223 = r83220 + r83222;
        double r83224 = sqrt(r83223);
        double r83225 = r83220 - r83222;
        double r83226 = sqrt(r83225);
        double r83227 = r83224 * r83226;
        double r83228 = r83220 + r83227;
        double r83229 = log(r83228);
        return r83229;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.2

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

  3. Applied add-sqr-sqrt32.2

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

  4. Applied difference-of-squares32.2

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

  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 2019308 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))