Average Error: 31.7 → 0.1
Time: 7.4s
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 r86958 = x;
        double r86959 = r86958 * r86958;
        double r86960 = 1.0;
        double r86961 = r86959 - r86960;
        double r86962 = sqrt(r86961);
        double r86963 = r86958 + r86962;
        double r86964 = log(r86963);
        return r86964;
}
↓
double f(double x) {
        double r86965 = x;
        double r86966 = 1.0;
        double r86967 = sqrt(r86966);
        double r86968 = r86965 + r86967;
        double r86969 = sqrt(r86968);
        double r86970 = r86965 - r86967;
        double r86971 = sqrt(r86970);
        double r86972 = r86969 * r86971;
        double r86973 = r86965 + r86972;
        double r86974 = log(r86973);
        return r86974;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  3. Applied add-sqr-sqrt31.7

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

  4. Applied difference-of-squares31.7

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

  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 2020064 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))