Average Error: 31.9 → 0.1
Time: 7.4s
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 r69024 = x;
        double r69025 = r69024 * r69024;
        double r69026 = 1.0;
        double r69027 = r69025 - r69026;
        double r69028 = sqrt(r69027);
        double r69029 = r69024 + r69028;
        double r69030 = log(r69029);
        return r69030;
}
↓
double f(double x) {
        double r69031 = x;
        double r69032 = 1.0;
        double r69033 = sqrt(r69032);
        double r69034 = r69031 + r69033;
        double r69035 = sqrt(r69034);
        double r69036 = r69031 - r69033;
        double r69037 = sqrt(r69036);
        double r69038 = r69035 * r69037;
        double r69039 = r69031 + r69038;
        double r69040 = log(r69039);
        return r69040;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

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

  3. Applied add-sqr-sqrt31.9

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

  4. Applied difference-of-squares31.9

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

  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 2019322 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))