Average Error: 32.0 → 0.0
Time: 8.6s
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 r28729 = x;
        double r28730 = r28729 * r28729;
        double r28731 = 1.0;
        double r28732 = r28730 - r28731;
        double r28733 = sqrt(r28732);
        double r28734 = r28729 + r28733;
        double r28735 = log(r28734);
        return r28735;
}
↓
double f(double x) {
        double r28736 = x;
        double r28737 = 1.0;
        double r28738 = sqrt(r28737);
        double r28739 = r28736 + r28738;
        double r28740 = sqrt(r28739);
        double r28741 = r28736 - r28738;
        double r28742 = sqrt(r28741);
        double r28743 = r28740 * r28742;
        double r28744 = r28736 + r28743;
        double r28745 = log(r28744);
        return r28745;
}

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-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))