Average Error: 32.0 → 0.1
Time: 2.4s
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 r66303 = x;
        double r66304 = r66303 * r66303;
        double r66305 = 1.0;
        double r66306 = r66304 - r66305;
        double r66307 = sqrt(r66306);
        double r66308 = r66303 + r66307;
        double r66309 = log(r66308);
        return r66309;
}
↓
double f(double x) {
        double r66310 = x;
        double r66311 = 1.0;
        double r66312 = sqrt(r66311);
        double r66313 = r66310 + r66312;
        double r66314 = sqrt(r66313);
        double r66315 = r66310 - r66312;
        double r66316 = sqrt(r66315);
        double r66317 = r66314 * r66316;
        double r66318 = r66310 + r66317;
        double r66319 = log(r66318);
        return r66319;
}

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 2020018 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))