Average Error: 32.0 → 0.1
Time: 7.3s
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 r48107 = x;
        double r48108 = r48107 * r48107;
        double r48109 = 1.0;
        double r48110 = r48108 - r48109;
        double r48111 = sqrt(r48110);
        double r48112 = r48107 + r48111;
        double r48113 = log(r48112);
        return r48113;
}
↓
double f(double x) {
        double r48114 = x;
        double r48115 = 1.0;
        double r48116 = sqrt(r48115);
        double r48117 = r48114 + r48116;
        double r48118 = sqrt(r48117);
        double r48119 = r48114 - r48116;
        double r48120 = sqrt(r48119);
        double r48121 = r48118 * r48120;
        double r48122 = r48114 + r48121;
        double r48123 = log(r48122);
        return r48123;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  3. Applied add-sqr-sqrt32.0

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

  4. Applied difference-of-squares32.0

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

  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 2019209 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))