Average Error: 31.8 → 0.1
Time: 5.0s
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 r65087 = x;
        double r65088 = r65087 * r65087;
        double r65089 = 1.0;
        double r65090 = r65088 - r65089;
        double r65091 = sqrt(r65090);
        double r65092 = r65087 + r65091;
        double r65093 = log(r65092);
        return r65093;
}
↓
double f(double x) {
        double r65094 = x;
        double r65095 = 1.0;
        double r65096 = sqrt(r65095);
        double r65097 = r65094 + r65096;
        double r65098 = sqrt(r65097);
        double r65099 = r65094 - r65096;
        double r65100 = sqrt(r65099);
        double r65101 = r65098 * r65100;
        double r65102 = r65094 + r65101;
        double r65103 = log(r65102);
        return r65103;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.8

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

  3. Applied add-sqr-sqrt31.8

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

  4. Applied difference-of-squares31.8

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

  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 2020001 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))