Average Error: 31.9 → 0.3
Time: 7.6s
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 r43273 = x;
        double r43274 = r43273 * r43273;
        double r43275 = 1.0;
        double r43276 = r43274 - r43275;
        double r43277 = sqrt(r43276);
        double r43278 = r43273 + r43277;
        double r43279 = log(r43278);
        return r43279;
}
↓
double f(double x) {
        double r43280 = x;
        double r43281 = 0.125;
        double r43282 = 3.0;
        double r43283 = pow(r43280, r43282);
        double r43284 = r43281 / r43283;
        double r43285 = 0.5;
        double r43286 = r43285 / r43280;
        double r43287 = r43284 + r43286;
        double r43288 = r43280 - r43287;
        double r43289 = r43280 + r43288;
        double r43290 = log(r43289);
        return r43290;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

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

  2. Taylor expanded around inf 0.3

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

  3. Simplified0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))