Average Error: 31.5 → 0.1
Time: 4.7s
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 r73159 = x;
        double r73160 = r73159 * r73159;
        double r73161 = 1.0;
        double r73162 = r73160 - r73161;
        double r73163 = sqrt(r73162);
        double r73164 = r73159 + r73163;
        double r73165 = log(r73164);
        return r73165;
}
↓
double f(double x) {
        double r73166 = x;
        double r73167 = 1.0;
        double r73168 = sqrt(r73167);
        double r73169 = r73166 + r73168;
        double r73170 = sqrt(r73169);
        double r73171 = r73166 - r73168;
        double r73172 = sqrt(r73171);
        double r73173 = r73170 * r73172;
        double r73174 = r73166 + r73173;
        double r73175 = log(r73174);
        return r73175;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  3. Applied add-sqr-sqrt31.5

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

  4. Applied difference-of-squares31.5

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

  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 2020056 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))