Average Error: 30.5 → 0.1
Time: 19.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)
double f(double x) {
        double r9620195 = x;
        double r9620196 = r9620195 * r9620195;
        double r9620197 = 1.0;
        double r9620198 = r9620196 - r9620197;
        double r9620199 = sqrt(r9620198);
        double r9620200 = r9620195 + r9620199;
        double r9620201 = log(r9620200);
        return r9620201;
}
↓
double f(double x) {
        double r9620202 = x;
        double r9620203 = 1.0;
        double r9620204 = r9620203 + r9620202;
        double r9620205 = sqrt(r9620204);
        double r9620206 = r9620202 - r9620203;
        double r9620207 = sqrt(r9620206);
        double r9620208 = r9620205 * r9620207;
        double r9620209 = r9620202 + r9620208;
        double r9620210 = log(r9620209);
        return r9620210;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.5

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

  3. Applied difference-of-sqr-130.5

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

  4. Applied sqrt-prod0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019125 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))