Average Error: 32.5 → 0.1
Time: 5.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{\sqrt{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{\sqrt{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)
double f(double x) {
        double r59391 = x;
        double r59392 = r59391 * r59391;
        double r59393 = 1.0;
        double r59394 = r59392 - r59393;
        double r59395 = sqrt(r59394);
        double r59396 = r59391 + r59395;
        double r59397 = log(r59396);
        return r59397;
}
↓
double f(double x) {
        double r59398 = x;
        double r59399 = 1.0;
        double r59400 = sqrt(r59399);
        double r59401 = r59398 + r59400;
        double r59402 = sqrt(r59401);
        double r59403 = sqrt(r59402);
        double r59404 = r59398 - r59400;
        double r59405 = sqrt(r59404);
        double r59406 = r59403 * r59405;
        double r59407 = r59403 * r59406;
        double r59408 = r59398 + r59407;
        double r59409 = log(r59408);
        return r59409;
}

Error

Bits error versus x

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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. Using strategy rm

  7. Applied add-sqr-sqrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Applied associate-*l*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019318 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))