Average Error: 32.2 → 0.4
Time: 9.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\left(\log 2 + \log \left(\sqrt{x}\right)\right) + \left(\left(\log \left(\sqrt{x}\right) - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\left(\log 2 + \log \left(\sqrt{x}\right)\right) + \left(\left(\log \left(\sqrt{x}\right) - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)
double f(double x) {
        double r98192 = x;
        double r98193 = r98192 * r98192;
        double r98194 = 1.0;
        double r98195 = r98193 - r98194;
        double r98196 = sqrt(r98195);
        double r98197 = r98192 + r98196;
        double r98198 = log(r98197);
        return r98198;
}
↓
double f(double x) {
        double r98199 = 2.0;
        double r98200 = log(r98199);
        double r98201 = x;
        double r98202 = sqrt(r98201);
        double r98203 = log(r98202);
        double r98204 = r98200 + r98203;
        double r98205 = 0.25;
        double r98206 = r98205 / r98201;
        double r98207 = r98206 / r98201;
        double r98208 = r98203 - r98207;
        double r98209 = 0.09375;
        double r98210 = 4.0;
        double r98211 = pow(r98201, r98210);
        double r98212 = r98209 / r98211;
        double r98213 = r98208 - r98212;
        double r98214 = r98204 + r98213;
        return r98214;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.2

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

  2. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(0.09375 \cdot \frac{1}{{x}^{4}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)}\]

  3. Simplified0.4

    \[\leadsto \color{blue}{\log 2 + \left(\left(\log x - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.4

    \[\leadsto \log 2 + \left(\left(\log \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)\]

  6. Applied log-prod0.4

    \[\leadsto \log 2 + \left(\left(\color{blue}{\left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{x}\right)\right)} - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)\]

  7. Applied associate--l+0.4

    \[\leadsto \log 2 + \left(\color{blue}{\left(\log \left(\sqrt{x}\right) + \left(\log \left(\sqrt{x}\right) - \frac{\frac{0.25}{x}}{x}\right)\right)} - \frac{0.09375}{{x}^{4}}\right)\]

  8. Applied associate--l+0.4

    \[\leadsto \log 2 + \color{blue}{\left(\log \left(\sqrt{x}\right) + \left(\left(\log \left(\sqrt{x}\right) - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)\right)}\]

  9. Applied associate-+r+0.4

    \[\leadsto \color{blue}{\left(\log 2 + \log \left(\sqrt{x}\right)\right) + \left(\left(\log \left(\sqrt{x}\right) - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)}\]

  10. Final simplification0.4

    \[\leadsto \left(\log 2 + \log \left(\sqrt{x}\right)\right) + \left(\left(\log \left(\sqrt{x}\right) - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))