Average Error: 32.1 → 0.0
Time: 9.2s
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 r44027 = x;
        double r44028 = r44027 * r44027;
        double r44029 = 1.0;
        double r44030 = r44028 - r44029;
        double r44031 = sqrt(r44030);
        double r44032 = r44027 + r44031;
        double r44033 = log(r44032);
        return r44033;
}
↓
double f(double x) {
        double r44034 = x;
        double r44035 = 1.0;
        double r44036 = sqrt(r44035);
        double r44037 = r44034 + r44036;
        double r44038 = sqrt(r44037);
        double r44039 = r44034 - r44036;
        double r44040 = sqrt(r44039);
        double r44041 = r44038 * r44040;
        double r44042 = r44034 + r44041;
        double r44043 = log(r44042);
        return r44043;
}

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-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))