Average Error: 32.5 → 0.1
Time: 2.7s
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 r92518 = x;
        double r92519 = r92518 * r92518;
        double r92520 = 1.0;
        double r92521 = r92519 - r92520;
        double r92522 = sqrt(r92521);
        double r92523 = r92518 + r92522;
        double r92524 = log(r92523);
        return r92524;
}
↓
double f(double x) {
        double r92525 = x;
        double r92526 = 1.0;
        double r92527 = sqrt(r92526);
        double r92528 = r92525 + r92527;
        double r92529 = sqrt(r92528);
        double r92530 = r92525 - r92527;
        double r92531 = sqrt(r92530);
        double r92532 = r92529 * r92531;
        double r92533 = r92525 + r92532;
        double r92534 = log(r92533);
        return r92534;
}

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 2019353 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))