Average Error: 31.6 → 0.0
Time: 11.8s
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 r1858112 = x;
        double r1858113 = r1858112 * r1858112;
        double r1858114 = 1.0;
        double r1858115 = r1858113 - r1858114;
        double r1858116 = sqrt(r1858115);
        double r1858117 = r1858112 + r1858116;
        double r1858118 = log(r1858117);
        return r1858118;
}
↓
double f(double x) {
        double r1858119 = x;
        double r1858120 = 1.0;
        double r1858121 = sqrt(r1858120);
        double r1858122 = r1858119 - r1858121;
        double r1858123 = sqrt(r1858122);
        double r1858124 = r1858119 + r1858121;
        double r1858125 = sqrt(r1858124);
        double r1858126 = r1858123 * r1858125;
        double r1858127 = r1858119 + r1858126;
        double r1858128 = log(r1858127);
        return r1858128;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.6

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

  3. Applied add-sqr-sqrt31.6

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

  4. Applied difference-of-squares31.6

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

  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 2019172 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))