Average Error: 32.6 → 0.2
Time: 6.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(x - \left(\frac{0.125}{{x}^{3}} + \frac{0.5}{x}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x - \left(\frac{0.125}{{x}^{3}} + \frac{0.5}{x}\right)\right)\right)
double f(double x) {
        double r25677 = x;
        double r25678 = r25677 * r25677;
        double r25679 = 1.0;
        double r25680 = r25678 - r25679;
        double r25681 = sqrt(r25680);
        double r25682 = r25677 + r25681;
        double r25683 = log(r25682);
        return r25683;
}
↓
double f(double x) {
        double r25684 = x;
        double r25685 = 0.125;
        double r25686 = 3.0;
        double r25687 = pow(r25684, r25686);
        double r25688 = r25685 / r25687;
        double r25689 = 0.5;
        double r25690 = r25689 / r25684;
        double r25691 = r25688 + r25690;
        double r25692 = r25684 - r25691;
        double r25693 = r25684 + r25692;
        double r25694 = log(r25693);
        return r25694;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019305 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))