Average Error: 32.8 → 0.1
Time: 2.3s
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 r63015 = x;
        double r63016 = r63015 * r63015;
        double r63017 = 1.0;
        double r63018 = r63016 - r63017;
        double r63019 = sqrt(r63018);
        double r63020 = r63015 + r63019;
        double r63021 = log(r63020);
        return r63021;
}
↓
double f(double x) {
        double r63022 = x;
        double r63023 = 1.0;
        double r63024 = sqrt(r63023);
        double r63025 = r63022 + r63024;
        double r63026 = sqrt(r63025);
        double r63027 = r63022 - r63024;
        double r63028 = sqrt(r63027);
        double r63029 = r63026 * r63028;
        double r63030 = r63022 + r63029;
        double r63031 = log(r63030);
        return r63031;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.8

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

  3. Applied add-sqr-sqrt32.8

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

  4. Applied difference-of-squares32.8

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

  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 2020020 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))