Average Error: 32.0 → 0.1
Time: 11.1s
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 r2296117 = x;
        double r2296118 = r2296117 * r2296117;
        double r2296119 = 1.0;
        double r2296120 = r2296118 - r2296119;
        double r2296121 = sqrt(r2296120);
        double r2296122 = r2296117 + r2296121;
        double r2296123 = log(r2296122);
        return r2296123;
}
↓
double f(double x) {
        double r2296124 = x;
        double r2296125 = 1.0;
        double r2296126 = sqrt(r2296125);
        double r2296127 = r2296124 - r2296126;
        double r2296128 = sqrt(r2296127);
        double r2296129 = r2296124 + r2296126;
        double r2296130 = sqrt(r2296129);
        double r2296131 = r2296128 * r2296130;
        double r2296132 = r2296124 + r2296131;
        double r2296133 = log(r2296132);
        return r2296133;
}

Error

Bits error versus x

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Results

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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 2019169 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))