Average Error: 32.3 → 0.1
Time: 2.6s
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 r72038 = x;
        double r72039 = r72038 * r72038;
        double r72040 = 1.0;
        double r72041 = r72039 - r72040;
        double r72042 = sqrt(r72041);
        double r72043 = r72038 + r72042;
        double r72044 = log(r72043);
        return r72044;
}
↓
double f(double x) {
        double r72045 = x;
        double r72046 = 1.0;
        double r72047 = sqrt(r72046);
        double r72048 = r72045 + r72047;
        double r72049 = sqrt(r72048);
        double r72050 = r72045 - r72047;
        double r72051 = sqrt(r72050);
        double r72052 = r72049 * r72051;
        double r72053 = r72045 + r72052;
        double r72054 = log(r72053);
        return r72054;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.3

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

  3. Applied add-sqr-sqrt32.3

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

  4. Applied difference-of-squares32.3

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

  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 2020039 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))