Average Error: 32.5 → 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 r1324087 = x;
        double r1324088 = r1324087 * r1324087;
        double r1324089 = 1.0;
        double r1324090 = r1324088 - r1324089;
        double r1324091 = sqrt(r1324090);
        double r1324092 = r1324087 + r1324091;
        double r1324093 = log(r1324092);
        return r1324093;
}
↓
double f(double x) {
        double r1324094 = x;
        double r1324095 = 1.0;
        double r1324096 = sqrt(r1324095);
        double r1324097 = r1324094 - r1324096;
        double r1324098 = sqrt(r1324097);
        double r1324099 = r1324094 + r1324096;
        double r1324100 = sqrt(r1324099);
        double r1324101 = r1324098 * r1324100;
        double r1324102 = r1324094 + r1324101;
        double r1324103 = log(r1324102);
        return r1324103;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.5

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

  3. Applied add-sqr-sqrt32.5

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

  4. Applied difference-of-squares32.5

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

  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 2019171 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))