Average Error: 32.0 → 0.1
Time: 9.8s
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 r3024024 = x;
        double r3024025 = r3024024 * r3024024;
        double r3024026 = 1.0;
        double r3024027 = r3024025 - r3024026;
        double r3024028 = sqrt(r3024027);
        double r3024029 = r3024024 + r3024028;
        double r3024030 = log(r3024029);
        return r3024030;
}
↓
double f(double x) {
        double r3024031 = x;
        double r3024032 = 1.0;
        double r3024033 = sqrt(r3024032);
        double r3024034 = r3024031 - r3024033;
        double r3024035 = sqrt(r3024034);
        double r3024036 = r3024031 + r3024033;
        double r3024037 = sqrt(r3024036);
        double r3024038 = r3024035 * r3024037;
        double r3024039 = r3024031 + r3024038;
        double r3024040 = log(r3024039);
        return r3024040;
}

Error

Bits error versus x

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Results

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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 2019174 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))