Average Error: 32.5 → 0.1
Time: 8.9s
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 r2326148 = x;
        double r2326149 = r2326148 * r2326148;
        double r2326150 = 1.0;
        double r2326151 = r2326149 - r2326150;
        double r2326152 = sqrt(r2326151);
        double r2326153 = r2326148 + r2326152;
        double r2326154 = log(r2326153);
        return r2326154;
}
↓
double f(double x) {
        double r2326155 = x;
        double r2326156 = 1.0;
        double r2326157 = sqrt(r2326156);
        double r2326158 = r2326155 - r2326157;
        double r2326159 = sqrt(r2326158);
        double r2326160 = r2326155 + r2326157;
        double r2326161 = sqrt(r2326160);
        double r2326162 = r2326159 * r2326161;
        double r2326163 = r2326155 + r2326162;
        double r2326164 = log(r2326163);
        return r2326164;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.5

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

  3. Applied add-sqr-sqrt32.5

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

  4. Applied difference-of-squares32.5

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

  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 2019171 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))