Average Error: 32.0 → 0.1
Time: 28.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1.0}\right)\]
↓
\[\log \left(x + \sqrt{x - \sqrt{1.0}} \cdot \sqrt{x + \sqrt{1.0}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1.0}\right)
↓
\log \left(x + \sqrt{x - \sqrt{1.0}} \cdot \sqrt{x + \sqrt{1.0}}\right)
double f(double x) {
        double r3545207 = x;
        double r3545208 = r3545207 * r3545207;
        double r3545209 = 1.0;
        double r3545210 = r3545208 - r3545209;
        double r3545211 = sqrt(r3545210);
        double r3545212 = r3545207 + r3545211;
        double r3545213 = log(r3545212);
        return r3545213;
}
↓
double f(double x) {
        double r3545214 = x;
        double r3545215 = 1.0;
        double r3545216 = sqrt(r3545215);
        double r3545217 = r3545214 - r3545216;
        double r3545218 = sqrt(r3545217);
        double r3545219 = r3545214 + r3545216;
        double r3545220 = sqrt(r3545219);
        double r3545221 = r3545218 * r3545220;
        double r3545222 = r3545214 + r3545221;
        double r3545223 = log(r3545222);
        return r3545223;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 32.0

    \[\log \left(x + \sqrt{x \cdot x - 1.0}\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt32.0

    \[\leadsto \log \left(x + \sqrt{x \cdot x - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\right)\]

  4. Applied difference-of-squares32.0

    \[\leadsto \log \left(x + \sqrt{\color{blue}{\left(x + \sqrt{1.0}\right) \cdot \left(x - \sqrt{1.0}\right)}}\right)\]

  5. Applied sqrt-prod0.1

    \[\leadsto \log \left(x + \color{blue}{\sqrt{x + \sqrt{1.0}} \cdot \sqrt{x - \sqrt{1.0}}}\right)\]

  6. Final simplification0.1

    \[\leadsto \log \left(x + \sqrt{x - \sqrt{1.0}} \cdot \sqrt{x + \sqrt{1.0}}\right)\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))