Average Error: 31.5 → 0.1
Time: 4.5s
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 r78961 = x;
        double r78962 = r78961 * r78961;
        double r78963 = 1.0;
        double r78964 = r78962 - r78963;
        double r78965 = sqrt(r78964);
        double r78966 = r78961 + r78965;
        double r78967 = log(r78966);
        return r78967;
}
↓
double f(double x) {
        double r78968 = x;
        double r78969 = 1.0;
        double r78970 = sqrt(r78969);
        double r78971 = r78968 + r78970;
        double r78972 = sqrt(r78971);
        double r78973 = r78968 - r78970;
        double r78974 = sqrt(r78973);
        double r78975 = r78972 * r78974;
        double r78976 = r78968 + r78975;
        double r78977 = log(r78976);
        return r78977;
}

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