Average Error: 32.1 → 0.2
Time: 9.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x - \frac{0.125}{{x}^{3}}\right) - \frac{0.5}{x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x - \frac{0.125}{{x}^{3}}\right) - \frac{0.5}{x}\right)\right)
double f(double x) {
        double r45430 = x;
        double r45431 = r45430 * r45430;
        double r45432 = 1.0;
        double r45433 = r45431 - r45432;
        double r45434 = sqrt(r45433);
        double r45435 = r45430 + r45434;
        double r45436 = log(r45435);
        return r45436;
}
↓
double f(double x) {
        double r45437 = x;
        double r45438 = 0.125;
        double r45439 = 3.0;
        double r45440 = pow(r45437, r45439);
        double r45441 = r45438 / r45440;
        double r45442 = r45437 - r45441;
        double r45443 = 0.5;
        double r45444 = r45443 / r45437;
        double r45445 = r45442 - r45444;
        double r45446 = r45437 + r45445;
        double r45447 = log(r45446);
        return r45447;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.1

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

  2. Taylor expanded around inf 0.2

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}\right)\]

  3. Simplified0.2

    \[\leadsto \log \left(x + \color{blue}{\left(\left(x - \frac{0.125}{{x}^{3}}\right) - \frac{0.5}{x}\right)}\right)\]

  4. Final simplification0.2

    \[\leadsto \log \left(x + \left(\left(x - \frac{0.125}{{x}^{3}}\right) - \frac{0.5}{x}\right)\right)\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))