Average Error: 32.0 → 0.1
Time: 3.1s
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 r71066 = x;
        double r71067 = r71066 * r71066;
        double r71068 = 1.0;
        double r71069 = r71067 - r71068;
        double r71070 = sqrt(r71069);
        double r71071 = r71066 + r71070;
        double r71072 = log(r71071);
        return r71072;
}
↓
double f(double x) {
        double r71073 = x;
        double r71074 = 1.0;
        double r71075 = sqrt(r71074);
        double r71076 = r71073 + r71075;
        double r71077 = sqrt(r71076);
        double r71078 = r71073 - r71075;
        double r71079 = sqrt(r71078);
        double r71080 = r71077 * r71079;
        double r71081 = r71073 + r71080;
        double r71082 = log(r71081);
        return r71082;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  3. Applied add-sqr-sqrt32.0

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

  4. Applied difference-of-squares32.0

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

  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 2019362 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))