Average Error: 32.1 → 0.2
Time: 9.1s
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 r49441 = x;
        double r49442 = r49441 * r49441;
        double r49443 = 1.0;
        double r49444 = r49442 - r49443;
        double r49445 = sqrt(r49444);
        double r49446 = r49441 + r49445;
        double r49447 = log(r49446);
        return r49447;
}
↓
double f(double x) {
        double r49448 = x;
        double r49449 = 0.125;
        double r49450 = 3.0;
        double r49451 = pow(r49448, r49450);
        double r49452 = r49449 / r49451;
        double r49453 = r49448 - r49452;
        double r49454 = 0.5;
        double r49455 = r49454 / r49448;
        double r49456 = r49453 - r49455;
        double r49457 = r49448 + r49456;
        double r49458 = log(r49457);
        return r49458;
}

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)))))