Average Error: 32.4 → 0.0
Time: 6.1s
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 r41988 = x;
        double r41989 = r41988 * r41988;
        double r41990 = 1.0;
        double r41991 = r41989 - r41990;
        double r41992 = sqrt(r41991);
        double r41993 = r41988 + r41992;
        double r41994 = log(r41993);
        return r41994;
}
↓
double f(double x) {
        double r41995 = x;
        double r41996 = 1.0;
        double r41997 = sqrt(r41996);
        double r41998 = r41995 + r41997;
        double r41999 = sqrt(r41998);
        double r42000 = r41995 - r41997;
        double r42001 = sqrt(r42000);
        double r42002 = r41999 * r42001;
        double r42003 = r41995 + r42002;
        double r42004 = log(r42003);
        return r42004;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 32.4

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

  3. Applied add-sqr-sqrt32.4

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

  4. Applied difference-of-squares32.4

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

  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 2019347 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))